TOPICS

Miscellaneous

Permutations and Combinations

**Question-1 :-** How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER ?

In the word DAUGHTER, there are 3 vowels namely, A, U, and E, and 5 consonants namely, D, G, H, T, and R. Number of ways of selecting 2 vowels out of 3 vowels =^{3}C_{2}= 3!/2!1! = 3 Number of ways of selecting 3 consonants out of 5 consonants =^{5}C_{3}= 5!/3!2! = 10 Therefore, number of combinations of 2 vowels and 3 consonants = 3 × 10 = 30 Each of these 30 combinations of 2 vowels and 3 consonants can be arranged among themselves in 5! ways. Hence, required number of different words = 30 × 5! = 3600

**Question-2 :-** 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?

In the word EQUATION, there are 5 vowels, namely, A, E, I, O, and U, and 3 consonants, namely, Q, T, and N. Since all the vowels and consonants have to occur together, both (AEIOU) and (QTN) can be assumed as single objects. Then, the permutations of these 2 objects taken all at a time are counted. This number would be ²P₂ Corresponding to each of these permutations, there are 5! permutations of the five vowels taken all at a time and 3! permutations of the 3 consonants taken all at a time. Hence, by multiplication principle, required number of words = 2! × 5! × 3! = 1440

**Question-3 :-** 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) atleast 3 girls ?

(iii) atmost 3 girls ?

A committee of 7 has to be formed from 9 boys and 4 girls. (i) Since exactly 3 girls are to be there in every committee, each committee must consist of (7 – 3) = 4 boys only. Thus, in this case, required number of ways =^{4}C_{3}x^{9}C_{4}= 4!/3!1! x 9!/4!5! = 504

(ii) Since at least 3 girls are to be there in every committee, the committee can consist of 3 girls and 4 boys can be selected in^{4}C_{3}x^{9}C_{4}ways. 4 girls and 3 boys can be selected in^{4}C_{4}x^{9}C_{3}ways . Therefore, in this case, required number of ways =^{4}C_{3}x^{9}C_{4}+^{4}C_{4}x^{9}C_{3}= 504 + 84 = 588

(iii) Since atmost 3 girls are to be there in every committee, the committee can consist of 3 girls and 4 boys can be selected in^{4}C_{3}x^{9}C_{4}ways. 2 girls and 5 boys can be selected in^{9}C_{2}x^{9}C_{5}ways. 1 girl and 6 boys can be selected in^{4}C_{1}x^{9}C_{6}ways. No girl and 7 boys can be selected in^{4}C_{0}x^{9}C_{7}ways. Therefore, in this case, required number of ways =^{4}C_{3}x^{9}C_{4}+^{4}C_{2}x^{9}C_{5}+^{4}C_{1}x^{9}C_{6}+^{4}C_{0}x^{9}C_{7}= 504 + 756 + 336 + 36 = 1632

**Question-4 :-** If the different permutations of all the letter 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 ?

In the given word EXAMINATION, there are 11 letters out of which, A, I, and N appear 2 times and all the other letters appear only once. The words that will be listed before the words starting with E in a dictionary will be the words that start with A only. Therefore, to get the number of words starting with A, the letter A is fixed at the extreme left position, and then the remaining 10 letters taken all at a time are rearranged. Since there are 2 Is and 2 Ns in the remaining 10 letters, Number of words starting with A = 10!/2!2! = 907200 Thus, the required numbers of words is 907200.

**Question-5 :-** 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 digit is repeated ?

A number is divisible by 10 if its units digits is 0. Therefore, 0 is fixed at the units place. Therefore, there will be as many ways as there are ways of filling 5 vacant places in succession by the remaining 5 digits (i.e., 1, 3, 5, 7 and 9). The 5 vacant places can be filled in 5! ways. Hence, required number of 6-digit numbers = 5! = 120

**Question-6 :-** 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 alphabet ?

2 different vowels and 2 different consonants are to be selected from the English alphabet. Since there are 5 vowels in the English alphabet, number of ways of selecting 2 different vowels from the alphabet =^{5}C_{2}= 5!/2!3! = 10 Since there are 21 consonants in the English alphabet, number of ways of selecting 2 different consonants from the alphabet =^{21}C_{2}= 21!/2!19! = 210 Therefore, number of combinations of 2 different vowels and 2 different consonants = 10 × 210 = 2100 Each of these 2100 combinations has 4 letters, which can be arranged among themselves in 4! ways. Therefore, required number of words = 2100 × 4! = 50400

**Question-7 :-** 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 ?

It is given that the question paper consists of 12 questions divided into two parts – Part I and Part II, containing 5 and 7 questions, respectively. A student has to attempt 8 questions, selecting at least 3 from each part. This can be done as follows. 3 questions from part I and 5 questions from part II can be selected in^{5}C_{3}x^{7}C_{5}ways. 4 questions from part I and 4 questions from part II can be selected in^{5}C_{4}x^{7}C_{4}ways. 5 questions from part I and 3 questions from part II can be selected in^{5}C_{5}x^{7}C_{3}ways. Thus, required number of ways of selecting questions =^{5}C_{3}x^{7}C_{5}+^{5}C_{4}x^{7}C_{4}+^{5}C_{5}x^{7}C_{3}= 210 + 175 + 35 = 420

**Question-8 :-** Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

From a deck of 52 cards, 5-card combinations have to be made in such a way that in each selection of 5 cards, there is exactly one king. In a deck of 52 cards, there are 4 kings. 1 king can be selected out of 4 kings in^{4}C_{1}ways. 4 cards out of the remaining 48 cards can be selected in^{48}C_{4}ways. Thus, the required number of 5-card combinations is^{4}C_{1}x^{48}C_{4}.

**Question-9 :-** 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 ?

5 men and 4 women are to be seated in a row such that the women occupy the even places. The 5 men can be seated in 5! ways. For each arrangement, the 4 women can be seated only at the cross marked places (so that women occupy the even places). M x M x M x M x M Therefore, the women can be seated in 4! ways. Thus, possible number of arrangements = 4! × 5! = 24 × 120 = 2880.

**Question-10 :-** 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 ?

From the class of 25 students, 10 are to be chosen for an excursion party. Since there are 3 students who decide that either all of them will join or none of them will join, there are two cases. Case I: All the three students join. Then, the remaining 7 students can be chosen from the remaining 22 students in^{22}C_{7}ways. Case II: None of the three students join. Then, 10 students can be chosen from the remaining 22 students in^{22}C_{10}ways. Thus, required number of ways of choosing the excursion party is^{22}C_{7}x^{22}C_{10}.

**Question-11 :-** In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are together ?

In the given word ASSASSINATION, the letter A appears 3 times, S appears 4 times, I appears 2 times, N appears 2 times, and all the other letters appear only once. Since all the words have to be arranged in such a way that all the Ss are together, SSSS is treated as a single object for the time being. This single object together with the remaining 9 objects will account for 10 objects. These 10 objects in which there are 3 As, 2 Is, and 2 Ns can be arranged in 10!/3!2!2! ways. Thus, required number of ways of arranging the letters of the given word 10!/3!2!2! = 151200

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