Question

4. (HHM Problem 2.1 #2) Assume that a vowel is one of the five letters A, E, I, O, or U. (a) How many eleven-letter sequences from the alphabet contain exactly three vowels? (b) How many of these have at least one repeated letter?

Answer 1

(a)

Number of ways of selecting 3 positions for vowels out of 11 is

C(11,3) = 165

Now these three vowels can be filled in 5*5*5 = 125 ways

And rest 11-3 = 8 letters can be filled by 26-5=21 consonanats in ways.

Number of ways of selecting 3 positions for vowels out of 11 is C(11,3) = 165 Now these three vowels can be filled in 5*5*5 = 125 ways And rest 11-3 = 8 letters can be filled by 26-5 = 21 consonanats in 21^8 ways. So possible number of 11 letter sequence is 165.125.21^8 Number of 11 letter sequence: 26^11 Number of 11 letter sequence with all letters different is: 26.25.24 ... 16 Number of 11 letter sequence with at least one repeated letter is: 26^11 - (26.25.24 .16)

So possible number of 11 letter sequence is

(b)

Number of 11 letter sequence : $26^{11}$

Number of 11 letter sequence with all letters different is : $26\cdot 25\cdot 24...\cdot 16$

Number of 11 letter sequence with at least one repeated letter is: $26^{11}-\left ( 26\cdot 25\cdot 24...\cdot 16 \right )$