### Permutations and Combinations

When given a number of objects and we would to find the possible different arrangements or selections , we may use **permutations** or **combinations**.

Permutations deal with **the arrangement** of objects. The order of the objects is **important**. The number of different arrangements of *r* objects from *n* distinct object is,

\(\\[20pt]\hspace{2em} ^{n}P_{r } \ = \ \displaystyle \frac{n!}{(n-r)!} \)

Combinations deal with **the selection** of objects. The order of the objects is **not important**. The number of different selections of *r* objects from *n* distinct object is,

\(\\[20pt]\hspace{2em} ^{n}C_{r } \ = \ \displaystyle \binom{n}{r} \ = \ \frac{n!}{r!(n-r)!} \)

Notice that it is also used as **the binomial coefficients** in Binomial Expansion and Binomial Series.

Let’s see a simple example below to illustrate the difference between permutations and combinations.

We have 3 letters, **A, B and C** and we would like to find the permutations and combinations of 2 letters from the 3 letters.

For permutations, we have the following arrangements: **AB, BA, AC, CA, BC and CB**.

For combinations, we have the following selections: **AB, AC and BC**.

As you can see, **AB and BA**, **AC and CA** or **BC and CB** are considered as the same selection since the order of the objects is not important.

In other words, combinations is actually the permutations of ** r identical objects** from

*n*distinct object.

Try the exercises below and if you need any help, just look at the solution I have written. Cheers ! =) .

\(\\[1pt]\)

EXERCISE 5A:

\(\\[1pt]\)

\({\small 2.\enspace}\) How many numbers are there between 1245 and 5421 inclusive which contain each of the digits 1, 2, 4 and 5 once and once only?

\(\\[1pt]\)

\({\small 3.\enspace}\) An artist is going to arrange five paintings in a row on a wall. In how many ways can this be done?

\(\\[1pt]\)

\({\small 4.\enspace}\) Ten athletes are running in a 100-metre race. In how many different ways can the first three places be filled?

\(\\[1pt]\)

\({\small 5.\enspace}\) By writing out all the possible arrangements of \({\small {D}_{1}{E}_{1}{E}_{2}{D}_{2} }\), show that there are \( \frac{4!}{2!(2)!} = {\small 6 \ }\) different arrangements of the letters of the word

*DEED*.

\(\\[1pt]\)

\({\small 6.\enspace}\) A typist has five letters and five addressed envelopes. In how many different ways can the letters be placed in each envelope without getting every letter in the right envelope? If the letters are placed in the envelopes at random what is the probability that each letter is in its correct envelope?

\(\\[1pt]\)

\({\small 7.\enspace}\) How many different arrangements can be made of the letters in the word

*STATISTICS*?

\(\\[1pt]\)

\({\small 8.\enspace \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) Calculate the number of arrangements of the letters in the word

*NUMBER*.

\(\\[1pt]\)

\({\small \hspace{2.3em}\textrm{(b)}.\hspace{0.8em}}\) How many of the arrrangements in part (a) begin and end with a vowel?

\(\\[1pt]\)

\({\small 9.\enspace}\) How many different numbers can be formed by taking one, two, three and four digits from the digits 1, 2, 7 and 8, if repetitions are not allowed?

One of these numbers is chosen at random. What is the probability that it is greater than 200?

\(\\[1pt]\)

EXERCISE 5B:

\(\\[1pt]\)

\({\small 2.\enspace}\) From a group of 30 boys and 32 girls, two girls and two boys are to be chosen to represent their school. How many possible selections are there?

\(\\[1pt]\)

\({\small 3.\enspace}\) A history exam paper contains eight questions, four in Part A and four in Part B. Candidates are required to attempt five questions. In how many ways can this be done if

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) there are no restrictions,

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) at least two questions from Part A and at least two questions from Part B must be attempted?

\(\\[1pt]\)

\({\small 4.\enspace}\) A committee of three people is to be selected from four women and five men. The rules state that there must be at least one man and one woman on the committee. In how many different ways can the committee be chosen?

Subsequently one of the men and one of the women marry each other. The rules also state that a married couple may not both serve on the committee. In how many ways can the committee be chosen now?

\(\\[1pt]\)

\({\small 5.\enspace}\) A box of one dozen eggs contains one that is bad. If three eggs are chosen at random, what is the probability that one of them will be bad?

\(\\[1pt]\)

\({\small 6.\enspace}\) In a game of bridge the pack of 52 cards is shared equally between all four players. What is the probability that one particular player has no hearts?

\(\\[1pt]\)

\({\small 7.\enspace}\) A bag contains 20 chocolates, 15 toffees and 12 peppermints. If three sweets are chosen at random, what is the probability that they are

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) all different,

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) all chocolates,

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(c)}.\hspace{0.8em}}\) all the same,

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(d)}.\hspace{0.8em}}\) all not chocolates?

\(\\[1pt]\)

\({\small 8.\enspace}\) Show that \( \displaystyle \binom{n}{r} \ = \ \binom{n}{n \ – \ r} \).

\(\\[1pt]\)

\({\small 9.\enspace}\) Show that the number of permutations of

*n*objects of which

*r*are of one kind and

*n – r*are of another kind is \( \displaystyle \binom{n}{r} \).

\(\\[1pt]\)

EXERCISE 5C:

*CONSTANTINOPLE*are written on 14 cards, one on each card. The cards are shuffled and then arranged in a straight line.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) How many different possible arrangements are there?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) How many arrangements begin with

*P*?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(c)}.\hspace{0.8em}}\) How many arrangements start and end with

*O*?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(d)}.\hspace{0.8em}}\) How many arrangements are there where no two vowels are next to each other?

\(\\[1pt]\)

\({\small 2.\enspace}\) A coin is tossed 10 times.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) How many different sequences of heads and tails are possible?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) How many different sequences containing six heads and four tails are possible?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(c)}.\hspace{0.8em}}\) What is the probability of getting six heads and four tails?

\(\\[1pt]\)

\({\small 3.\enspace}\) Eight cards are selected with replacement from a standard pack of 52 playing cards, with 12 picture cards, 20 odd cards and 20 even cards.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) How many different sequences of eight cards are possible?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) How many of the sequences in part (a) will contain three picture cards, three odd-numbered cards and two even-numbered cards?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(c)}.\hspace{0.8em}}\) Use parts (a) and (b) to determine the probability of getting three picture cards, three odd-numbered cards and two even-numbered cards if eight cards are selected with replacement from a standard pack of 52 playing cards.

\(\\[1pt]\)

\({\small 4.\enspace}\) Eight women and five men are standing in a line.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) How many arrangements are possible if any individual can stand in any position?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) In how many arrangements will all five men be standing next to one another?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(c)}.\hspace{0.8em}}\) In how many arrangements will no two men be standing next to one another?

\(\\[1pt]\)

\({\small 5.\enspace}\) Each of the digits 1, 1, 2, 3, 3, 4, 6 is written on a separate card. The seven cards are then laid out in a row to form a 7-digit number.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) How many distinct 7-digit numbers are there?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) How many of these 7-digit numbers are even?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(c)}.\hspace{0.8em}}\) How many of these 7-digit numbers are divisible by 4?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(d)}.\hspace{0.8em}}\) How many of these 7-digit numbers start and end with the same digit?

\(\\[1pt]\)

\({\small 6.\enspace}\) Three families, the Mehtas, the Mupondas and the Lams, go to the cinema together to watch a film. Mr and Mrs Mehta take their daughter Indira, Mr and Mrs Muponda take their sons Paul and John, and Mrs Lam takes her children Susi, Kim and Lee. The families occupy a single row with eleven seats.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) In how many ways could the eleven people be seated if there were no restriction?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) In how many ways could the eleven people sit down so that the members of each family are all sitting together?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(c)}.\hspace{0.8em}}\) In how many of the arrangements will no two adults be sitting next to one another?

\(\\[1pt]\)

\({\small 7.\enspace}\) The letters of the word

*POSSESSES*are written on nine cards, one on each card. The cards are shuffled and four of them are selected and arranged in a straight line.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) How many possible selections are there of four letters?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) How many arrangements are there of four letters?

\(\\[1pt]\)

MISCELLANEOUS EXERCISE 5:

\(\\[1pt]\)

\({\small 2.\enspace}\) In how many ways can a committee of four men and four women be seated in a row if

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) they can sit in any position,

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) no one is seated next to a person of the same sex?

\(\\[1pt]\)

\({\small 3.\enspace}\) How many distinct arrangements are there of the letters in the word

*ABRACADABRA*?

\(\\[1pt]\)

\({\small 4.\enspace}\) Six people are going to travel in a six-seater minibus but only three of them can drive. In how many different ways can they seat themselves?

\(\\[1pt]\)

\({\small 5.\enspace}\) There are eight different books on a bookshelf: three of them are hardbacks and the rest are paperbacks.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) In how many different ways can the books be arranged if all the paperbacks are together and all the hardbacks are together?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) In how many different ways can the books be arranged if all the paperbacks are together?

\(\\[1pt]\)

\({\small 6.\enspace}\) Four boys and two girls sit in a line on stools in front of a coffee bar.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) In how many ways can they arrange themselves so that the two girls are together?

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) In how many ways can they sit if the two girls are not together?

\(\\[1pt]\)

\({\small 7.\enspace}\) Ten people travel in two cars, a saloon and a Mini. If the saloon has seats for six and the Mini has seats for four, find the number of different ways in which the party can travel, assuming that the order of seating in each car does not matter and all the people can drive.

\(\\[1pt]\)

\({\small 8.\enspace}\) Giving a brief explanation of your method, calculate the number of different ways in which the letters of the word

*TRIANGLES*can be arranged if no two vowels may come together.

\(\\[1pt]\)

\({\small 9.\enspace}\) I have seven fruit bars to last the week. Two are apricot, three fig and two peach. I select one bar each day. In how many different orders can I eat the bars?

If I select a fruit bar at random each day, what is the probability that I eat the two appricot ones on consecutive days?

\(\\[1pt]\)

\({\small 10.\enspace}\) A class contains 30 children, 18 girls and 12 boys. Four complimentary theatre tickets are distributed at random to the children in the class. What is the probability that

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) all four tickets go to the girls,

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) two boys and two girls receive tickets?

\(\\[1pt]\)

\({\small 11.\enspace \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) How many different 7-digit numbers can be formed from the digits 0, 1, 2, 2, 3, 3, 3 assuming that a number cannot start with 0?

\(\\[1pt]\)

\({\small \hspace{2.8em}\textrm{(b)}.\hspace{0.8em}}\) How many of these numbers will end in 0?

\(\\[1pt]\)

\({\small 12.\enspace}\) Calculate the number of ways in which three girls and four boys can be seated on a row of seven chairs if each arrangement is to be symmetrical.

\(\\[1pt]\)

\({\small 13.\enspace}\) Find the number of ways in which

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) 3 people can be arranged in 4 seats,

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) 5 people can be arranged in 5 seats.

\(\\[1pt]\)

In a block of 8 seats, 4 are in row A and 4 are in row B. Find the number of ways of arranging 8 people in the 8 seats given that 3 specified people must be in row A.

\(\\[1pt]\)

\({\small 14.\enspace}\) Eight different cards, of which four are red and four are black, are dealt to two players so that each receives a hand of four cards. Calculate

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) the total number of different hands which a given player could receive,

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) the probability that each player receives a hand consisting of four cards all of the same colour.

\(\\[1pt]\)

\({\small 15.\enspace}\) A piece of wood of length of 10 cm is to be divided into 3 pieces so that the length of each piece is a whole number of cm, for example, 2 cm, 3 cm and 5 cm.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) List all the different sets of lengths which could be obtained.

\(\\[1pt]\)

\({\small \hspace{1.2em}\textrm{(b)}.\hspace{0.8em}}\) If one of these sets is selected at random, what is the probability that the lengths of the pieces could be lengths of the sides of a triangle?

\(\\[1pt]\)

\({\small 16.\enspace}\) Nine persons are to be seated at three tables holding 2, 3 and 4 persons respectively. In how many ways can the groups sitting at the tables be selected, assuming that the order of sitting at the tables does not matter?

\(\\[1pt]\)

\({\small 17.\enspace \hspace{1.2em}\textrm{(a)}.\hspace{0.8em}}\) Calculate the number of different arrangements which can be made using all the letters of the word

*BANANA*.

\(\\[1pt]\)

\({\small \hspace{2.8em}\textrm{(b)}.\hspace{0.8em}}\) The number of combinations of 2 objects from

*n*is equal to the number of combinations of 3 objects from

*n*. Determine

*n*.

\(\\[1pt]\)

\({\small 18.\enspace}\) A ‘hand’ of 5 cards is dealt from an ordinary pack of 52 playing cards. Show that there are nearly 2.6 million distinct hands and that, of these, 575 757 contain no card from the heart suit.

On three successive occasions a card player is dealt a hand containing no heart. What is the probability of this happening? What conclusion might the player be justifiably reach?

\(\\[1pt]\)

PRACTICE MORE WITH THESE QUESTIONS BELOW!

\({\small 1.\enspace}\) In a 60-metre hurdles race there are five runners, one from each of the nations Austria, Belgium, Canada, Denmark and England.

\({\small\hspace{1.2em} (\textrm{i}).\hspace{0.8em}}\) How many different finishing orders are there?

\({\small\hspace{1.2em} (\textrm{ii}).\hspace{0.7em}}\) What is the probability of predicting the finishing order by choosing first, second, third, fourth and fifth at random?

\({\small 2. \enspace}\) In a ‘Goal of the season’ competition, participants are asked to rank ten goals in order of quality.

The organisers select their ‘correct’ order at random. Anybody who matches their order will be invited to join the television commentary team for the next international match.

\({\small\hspace{1.2em} (\textrm{i}).\hspace{0.8em}}\) What is the probability of a participant’s order being the same as that of the organisers?

\({\small\hspace{1.2em} (\textrm{ii}).\hspace{0.7em}}\) Five million people enter the competition. How many people would be expected to join the commentary team?

\({\small 3. \enspace}\) How many arrangements of the word ACHIEVE are there if

\({\small\hspace{1.2em} (\textrm{i}).\hspace{0.8em}}\) there are no restrictions on the order the letters are to be in

\({\small\hspace{1.2em} (\textrm{ii}).\hspace{0.7em}}\) the first letter is an A

\({\small\hspace{1.2em} (\textrm{iii}).\hspace{0.5em}}\) the letters A and I are to be together.

\({\small\hspace{1.2em} (\textrm{iv}).\hspace{0.6em}}\)the letters C and H are to be apart.

\({\small 4. \enspace (\textrm{i}).\hspace{0.7em}}\) A football team consists of 3 players who play in a defence position, 3 players who play in a midfield position and 5 players who play in a forward position. Three players are chosen to collect a gold medal for the team. Find in how many ways this can be done

\({\small\hspace{2.8em}(\textrm{a}).\hspace{0.7em}}\) if the captain, who is a midfield player, must be included, together with one defence and one forward player.

\({\small\hspace{2.8em}(\textrm{b}).\hspace{0.7em}}\) if exactly one forward player must be included, together with any two others.

\({\small \hspace{1.2em} (\textrm{ii}).\hspace{0.6em}}\) Find how many different arrangements there are of the nine letters in the words GOLD MEDAL

\({\small\hspace{2.8em}(\textrm{a}).\hspace{0.7em}}\) if there are no restrictions on the order of the letters,

\({\small\hspace{2.8em}(\textrm{b}).\hspace{0.7em}}\) if the two letters D come first and the letters L come last.

\({\small 5. \enspace}\) The diagram shows the seating plan for passengers in a minibus, which has 17 seats arranged in 4 rows. The back row has 5 seats and the other 3 rows have 2 seats on each side. 11 passengers get on the minibus.

\(\\[1pt]\)

\(\\[1pt]\)

\({\small\hspace{1.2em} (\textrm{i}).\hspace{0.7em}}\) How many possible seating arrangements are there for the 11 passengers?

\({\small\hspace{1.2em} (\textrm{ii}).\hspace{0.6em}}\) How many possible seating arrangements are there if 5 particular people sit in the back row?

Of the 11 passengers, 5 are unmarried and the other 6 consist of 3 married couples.

\({\small\hspace{1.2em} (\textrm{iii}).\hspace{0.5em}}\) In how many ways can 5 of the 11 passengers on the bus be chosen if there must be 2 married couples and 1 other person, who may or may not be married?

\({\small 6. \enspace}\) A choir consists of 13 sopranos, 12 altos, 6 tenors and 7 basses. A group consisting of 10 sopranos, 9 altos, 4 tenors and 4 basses is to be chosen from the choir.

\({\small\hspace{1.2em} (\textrm{i}).\hspace{0.7em}}\) In how many different ways can the group be chosen?

\({\small\hspace{1.2em} (\textrm{ii}).\hspace{0.6em}}\) In how many ways can the 10 chosen sopranos be arranged in a line if the 6 tallest stand next to each other?

\({\small\hspace{1.2em} (\textrm{iii}).\hspace{0.5em}}\) The 4 tenors and the 4 basses in the group stand in a single line with all the tenors next to each other and all the basses next to each other. How many possible arrangements are there if three of the tenors refuse to stand next to any of the basses?

\({\small 7. \enspace (\textrm{i}).\hspace{0.7em}}\) Find how many numbers between 5000 and 6000 can be formed from the digits 1, 2, 3, 4, 5 and 6

\({\small\hspace{2.8em}(\textrm{a}).\hspace{0.7em}}\) if no digits are repeated,

\({\small\hspace{2.8em}(\textrm{b}).\hspace{0.7em}}\) if repeated digits are allowed.

\({\small \hspace{1.2em} (\textrm{ii}).\hspace{0.6em}}\) Find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls

\({\small\hspace{2.8em}(\textrm{a}).\hspace{0.7em}}\) if there are more girls than boys in the team,

\({\small\hspace{2.8em}(\textrm{b}).\hspace{0.7em}}\) if three of the boys are cousins and are either all in the team or all not in the team.

\({\small 8. \enspace (\textrm{i}).\hspace{0.7em}}\) Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that all four Es are together.

\({\small \hspace{1.2em} (\textrm{ii}).\hspace{0.6em}}\) Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that the Ss are not next to each other.

Four letters are selected from the 12 letters of the word STEEPLECHASE.

\({\small \hspace{1.2em} (\textrm{iii}).\hspace{0.4em}}\) Find the number of different selections if the four letters include exactly one S.

\({\small 9. \enspace (\textrm{i}).\hspace{0.7em}}\) Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that all three Os are together and both Ts are together.

\({\small \hspace{1.2em} (\textrm{ii}).\hspace{0.6em}}\) Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that the Ts are not together.

\({\small \hspace{1.2em} (\textrm{iii}).\hspace{0.4em}}\) Find the probability that a randomly chosen arrangement of the 9 letters of the word TOADSTOOL has a T at the beginning and a T at the end.

\({\small \hspace{1.2em} (\textrm{iv}).\hspace{0.6em}}\) Five letters are selected from the 9 letters of the word TOADSTOOL. Find the number of different selections if the five letters include at least 2 Os and at least 1 T.

\({\small 10. \enspace (\textrm{a}).\hspace{0.7em}}\) A group of 6 teenagers go boating. There are three boats available. One boat has room for 3 people, one has room for 2 people and one has room for 1 person. Find the number of different ways the group of 6 teenagers can be divided between the three boats.

\({\small \hspace{1.6em} (\textrm{b}).\hspace{0.6em}}\) Find the number of different 7-digit numbers which can be formed from the seven digits 2, 2, 3, 7, 7, 7, 8 in each of the following cases.

\({\small \hspace{2.8em} (\textrm{i}).\hspace{0.7em}}\) The odd digits are together and the even digits are together.

\({\small \hspace{2.8em} (\textrm{ii}).\hspace{0.6em}}\) The 2s are not together.

As always, if you have any particular questions to discuss, leave it in the comment section below. Cheers =) .