# Permutations and Combinations

**Exercise 1**

**1. **How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that:

(i) repetition of the digits is allowed

(ii) repetition of the digits is not allowed.

**2. ** How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?

**3. ** How many four-letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?

**4. ** How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?

**5. **A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?

**6. ** Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?

**Exercise 2**

**7. **Evaluate:

(i) 8!

(ii) 4! – 3!

**8. **Is 3! + 4! = 7!?

**Exercise 3**

**12.** How many 3-digit numbers can be formed by using the digits 1 to 9 if no digits is repeated?

**13.** How many 4-digit numbers are there with no digit repeated?

**14.** How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7 if no digit is repeated?

**15.** Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?

**16.** From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person cannot hold more than one position?

**17. **Find n if^{(n-1)} P_{3}:^{n} P_{3} = 1 : 9

**18.** Find r if:

(i) ^{5}P_{r} = 2^{6 }p_{r-1}

(ii) ^{5}P_{r} = 6P_{r-1}

**19.** How many words, with or without meaning can be formed using all the letters of the word EQUATION, using each letter exactly once?

**20. **How many words, with or without meaning can be formed using all the letters of the word MONDAY, assuming that no letter is repeated if:

(i) 4 letters are used at a time

(ii) all letters are used at a time

(iii) all letters are used by first letter is a vowel?

**21.** In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together?

**22. **In how many ways the letters of word PERMUTATIONS be arranged if the

(i) words start with P and end with S.

(ii) vowels are together.

(iii) there are always 4 letters between P and S.

**Exercise 4**

**23.** If ^{n}C_{8} = ^{n}C_{2}, find ^{n}C_{2}

**24.** Determine if:

(i) ^{n}C_{3} : ^{n}C_{2} = 12 : 1

(ii) ^{n}C_{3} : ^{n}C_{3} = 11 : 1

**25.** How many chords can be drawn through 21 points on a circle?

**26. **In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

**27.** Find the number ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

**28.** Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

**29.** In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?

**30.** A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

**31.** In how many ways can a student choose a program of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

**Exercise 5**

**32.** How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?

**33. ** How many words, with or without meaning can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?

**34. ** A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of:

(i) exactly 3 girls

(ii) at least 3 girls

(iii) almost 3 girls?

**35.** If the different permutations of all the letters of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?

**36.** How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digits are repeated?

**37.** The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabets?

**38.** In an examination, a question paper consists of 12 questions divided into two parts i.e., part I and part II containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

**39.** Determine the number of 5-card combinations out of a deck of 52 cards is each selection of 5 cards has exactly one king.

**40.** It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?

**41.** From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

**42.** In how many ways can be letters of the word ASSASSINATION be arranged so that all the S’s are together?