Combinatorics Questions

Biju Ale | Mar 28, 2021

Mathematics

Independent Events

You have three shirts and four pairs of pants. How many outfits consisting of one shirt and one pair of pants can you make?
In how many ways can a pair of die appear when rolled?
In how many ways can a coin come up if flipped 7 times?
In how many ways can 4 coins and 3 dies come up if flipped together?
In how many ways a four-lettered word can be formed in English?
In how many ways can Horatio give M&M’s to three kids? (M&M’s come in red, yellow, green, brown, and tan.)
In how many ways four-lettered words can be formed with vowels in the middle two places & consonants at the end?
How many odd numbers with the third digit 5 are there between 20000 and 69999 inclusive?
In how many ways can we form a license plate if there are 7 characters, none of which is the letter O, the first of which is a numerical digit (0-9), the second of which is a letter, and the remaining five of which can be either a digit or a letter (but not the letter O)?

Dependent Events

In how many ways can Michelangelo give five differently-colored M&M’s to five kids?
How many odd numbers with the middle digit 5 are there between 40000 and 69999 inclusive, with no digits repeated?
How many four-letter words can be made without repeating the same letter twice? (You can leave your answer as a product of integers.)
In how many ways can a five-letter word be written using only the first half of the alphabet with no repetitions such that the third and fifth letters are vowels and the first a consonant?
How many four-letter words are there with vowels in the middle two places and consonants in the other two, and with no letter repeated?
You have three shirts and four pairs of pants. How many outfits consisting of one shirt and one pair of pants can you make?
In how many ways can we form an international commission if we must choose one European country from among 6 European countries, one Asian country from among 4, one North American country from among 3, and one African country from among 7?
Your math club has 20 members. In how many ways can it select a president, a vice-president, and a treasurer if no member can hold more than one office?

Independent Events (needing casework)

How many odd numbers with the middle digit 5 and no digit repeated are there between 20000 and 69999?
On the island of Mumble, the Mumblian alphabet has only 5 letters, and every word in the Mumblian language has no more than 3 letters in it. How many words are possible? (A word can use a letter more than once, but 0 letters does not count as a word.)
The inhabitants of the island of Mumble decided to expand their alphabet to the standard Roman alphabet (26 letters, A through Z). But they still limit their words to 3 letters or less, and for some reason, they insist that all words contain the letter A at least once. How many 3-letter words are possible?

Arrangements

In how many ways can I arrange four different books on a shelf?
In how many ways can we arrange n different books, where n is a positive integer?
How many distinct arrangements are there for the following words: DOG, BALL, TATTER, PAPA

Arrangements with Restrictions

In how many ways can the word HELLO be made if two Ls are paired?
In how many ways can the letters PARALLEL be formed when P & R are together?
In how many ways can 4 women and 3 men stand in a row if the 3 men stand together
?In how many ways can 2 men and 3 women sit on a bench if the men must sit on the ends (1 on right end another on left end)?
In how many ways can 3 blue books and 4 red books be arranged on a shelf if a red book must be on each of the ends assuming that each book looks different except for colors?
In how many ways can the letters SUCCESS be arranged if no two S’s are next to one another?

Arrangements where order doesn't matter (Combination)

Answer each
1. In how many ways can a President and a Vice-President be chosen from a group of 4 people (assuming that the President and the Vice-President cannot be the same person)?
2. Another choice is made but without assigning roles, In how many ways can a 2-person committee be chosen from a group of 4 people (where the order in which we choose the 2 people doesn’t matter)?
3. What is the difference between the two scenarios in a, and in b? In how many ways can a President, Vice-President and Treasurer be chosen from a group of 8 people? Given three members of the group — Alice, Bob, and Carol - in how many ways can these three be chosen to be the three officers?
In how many ways can a 3-person committee be chosen from a group of 8 people?
In my state’s lottery, 48 balls are numbered from 1 to 48, and 6 are chosen. How many different sets of winning numbers are there? (In this lottery, the order in which the numbers are chosen does not matter.)

Some A-Level (CIE) Questions

9709/62 Feb/Mar 2016Hannah chooses 5 singers from 15 applicants to appear in a concert. She lists the 5 singers in the order in which they will perform. (i) How many different lists can Hannah make? [2] Of the 15 applicants, 10 are female and 5 are male. (ii) Find the number of lists in which the first performer is male, the second is female, the third is male, the fourth is female and the fifth is male. [2] Hannah’s friend Ami would like the group of 5 performers to include more males than females. The order in which they perform is no longer relevant. (iii) Find the number of different selections of 5 performers with more males than females. 3 Two of the applicants are Mr. and Mrs. Blake. Find the number of different selections that include Mr and Mrs. Blake and also fulfill Ami’s requirement. [3]
9709/62 Feb/Mar 2017 (i)A plate of cakes holds 12 different cakes. Find the number of ways these cakes can be shared between Alex and James if each receives an odd number of cakes. [3] (ii) Another plate holds 7 cupcakes, each with a different color icing, and 4 brownies, each of adifferent sizes. Find the number of different ways these 11 cakes can be arranged in a row if nobrownie is next to another brownie.[3] (iii) A plate of biscuits holds 4 identical chocolate biscuits, 6 identical shortbread biscuits, and 2 identical gingerbread biscuits. These biscuits are all placed in a row. Find how many different arrangements are possible if the chocolate biscuits are all kept together. [3]
9709/62 Feb/Mar 2018 The digits 1, 3, 5, 6, 6, 6, 8 can be arranged to form many different 7-digit numbers. (i) How many of the 7-digit numbers have all the even digits together and all the odd digits together? 3 How many of the 7-digit numbers are even? [3]
9709/62 Feb/Mar 2019 Find the number of different arrangements that can be made of all 9 letters in the word CAMERAMAN in each of the following cases. (i) There are no restrictions. [2] (ii) The As occupying the 1st, 5th and 9th positions. [1] (iii) There is exactly one letter between Ms. [4] Three letters are selected from the 9 letters of the word CAMERAMAN. (iv) Find the number of different selections if the three letters include exactly one M and exactly one A.[1] (v) Find the number of different selections if the three letters include at least one M. [3]
9709/61 Feb/Mar 2019 Freddie has 6 toy cars and 3 toy buses, all different. He chooses 4 toys to take on holiday with him.(i) In how many different ways can Freddie choose 4 toys? [1] (ii) How many of these choices will include both his favourite car and his favourite bus? [2] Freddie arranges these 9 toys in a line.(iii) Find the number of possible arrangements if the buses are all next to each other. [3] (iv) Find the number of possible arrangements if there is a car at each end of the line and no buses are next to each other. [3]
9709/61 Oct/Nov 2018 In an orchestra, there are 11 violinists, 5 cellists, and 4 double bass players. A small group of 6 musicians is to be selected from these 20. (i) How many different selections of 6 musicians can be made if there must be at least 4 violinists, at least 1 cellist, and no more than 1 double bass player?[4] The small group that is selected contains 4 violinists, 1 cellist, and 1 double bass player. They sit in a line to perform a concert. (ii) How many different arrangements are there of these 6 musicians if the violinists must sit together?
9709/62 May/Jun 2017 A library contains 4 identical copies of book A, 2 identical copies of book B and 5 identical copies of book C. These 11 books are arranged on a shelf in the library. (i) Calculate the number of different arrangements if the end books are either both book A or both book B. 4 Calculate the number of different arrangements if all the books A are next to each other and none of the books B are next to each other. [5]
9709/61 May/June 2017 Eight children of different ages stand in a random order in a line. Find the number of different ways this can be done if none of the three youngest children stand next to each other. [3]

David chooses 5 chocolates from 6 different dark chocolates, 4 different white chocolates, and 1 milk chocolate. He must choose at least one of each type. Find the number of different selections he can make. [4]

A password for Chelsea’s computer consists of 4 characters in a particular order. The characters are chosen from the following. The 26 capital letters A to Z The 9 digits 1 to 9 The 5 symbols # ~ * ? ! The password must include at least one capital letter, at least one digit, and at least one symbol. No character can be repeated. Find the number of different passwords that Chelsea can make. [4]

Answers