Chapter 6 - Section 6.3 - Permutations and Combinations - Exercises - Page 414: 31

a) 122,523,030 b) 72,930,375 c) 223,149,655 d) 100,626,625

a) We need to choose the position for the vowel which can be done in 6 ways. Next we need to choose the vowel to use which can be done in 5 ways. Each of the other five positions in the string can contain any of the 21 consonants. So there are $ {21}^5$ ways to fill these other five positions. Therefore, the answer is $$ 6 . 5 . {21}^5 = 122,523,030$$ b) First, we need to choose the position of the two vowels out of six which can be done in $^6C_2 = 15$ ways (we need to choose two positions out of six). We need to choose the two vowels ({ 5}^2 ways). Each of the other four positions can be filled by any 21 consonants. So, there are ${21}^4$ ways to fill these positions. Therefore, the answer is $$15 . {5}^2 . {21} ^4 = 72,930,375.$$ c) Number of strings having atleast one vowel= Total number of strings - Number of strings with no vowels. Total number of strings : Since for each position we have 26 choices, we obtain ${26}^6$ ways Number of strings with no vowel: Since for each position we have 26-5 choices, we obtain ${21}^6$ ways Therefore, the answer is $${26}^6- {21}^6 = 223,149,655.$$ d) Number of strings having atleast two vowel= Total number of strings - (Number of strings with no vowels + Number of strings with one vowels). Total number of strings : Since for each position we have 26 choices, we obtain ${26}^6$ ways Number of strings with no vowel: Since for each position we have 26-5 choices, we obtain ${21}^6$ ways Number of strings with one vowel: By part a), 6 . 5 . {21}^5 ways Therefore, the answer is $${26}^6- ({21}^6 + 6 . 5 . {21}^5) = 223149655 - 122523030 = 100,626,625$$