Class XI Chapter 7 – Permutation and Combination Maths ______________________________________________________________________________ Question 9: In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student? Solution 9: There are 9 courses available out of which, 2 specific courses are compulsory for every student. Therefore, every student has to choose 3 courses out of the remaining 7 courses. This can be chosen in 7 C 3 ways. Thus, required number of ways of choosing the programme 7! 765 4! 3!4! 3 21 4! 7 C3 35 . Miscellaneous Exercise 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? Solution 1: 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 of 3 vowels 3 C 2 3 Number of ways of selecting 3 consonants out of 5 consonants 5 C 3 10 310 30 Therefore, number of combinations of 2 vowels and 3 consonants 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 . 35! 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? Solution 2: 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 Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

Class XI Chapter 7 – Permutation and Combination Maths ______________________________________________________________________________ assumed as single objects. Then, the permutations of these 2 objects taken all at a time are counted. This number would be 2 P2 2! 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? Solution 3: (i) A committee of 7 has to be formed from 9 boys and 4 girls. 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 98 7 65! 4 43 25! 4 9 4! 9! C3 C4 3!1! 4!5! 504 (ii) Since at least 3 girls are to be there in every committee, the committee can consist of (a) 3 girls and 4 boys or (b) 4 girls and 3 boys 3 girls and 4 boys can be selected in 4 girls and 3 boys can be selected in C C 4 9 3 4 C C 4 9 4 3 Therefore, in this case, required number of ways ways. ways. C C C C 4 9 4 9 3 4 4 3 504 84 588 (iii) Since atmost 3 girls are to be there in every committee, the committee can consist of (a) 3 girls and 4 boys (b) 2 girls and 5 boys (c) 1 girl and 6 boys (d) No girl and 7 boys 3 girls and 4 boys can be selected in 4 C 9 C ways. 3 4 2 girls and 5 boys can be selected in 4 C 9 C ways. 1 girl and 6 boys can be selected in C 2 5 C 4 9 1 6 ways. No girl and 7 boys can be selected in 4 C 9 C ways. 0 7 Therefore, in this case, required number of ways Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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