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The english alphabet contains 21 consonants and five vowels.How many strings of six lowercase letters of the English alphabet contain

(b)Exactly two vowels

(d)At least two vowels

For (b) part I solved it like choose 2 vowels from 5 in $\binom{5}{2}$ ways and remaining 4 consonants in $\binom{21}{4}$ ways and permute this string in 6! ways. So total -> $\binom{5}{2}$ * $\binom{21}{4}$ *6!

and for (d) part

I did it by complimentary counting and subtracted case of having atmost 1 vowel from all combinations of strings of 6 letters.

and this was

$26^6$ - $(21^{6} - (\binom{5}{1}*\binom{21}{5}*6!))$

and both of my answers don't match with the key in Rosen. Please let me know where I am wrong.
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It is considering repetitions to be allowed. In that case,

For b)

Two characters have 5 options(vowels) = 52

4 characters have 21 options(consonants) = 214

They can be arranged in 6!/(4!*2!)

Thus, the final answer = 52 * 214 * 6!/(4!*2!)

For d)

Total number of arrangements - (number of arrangements with no vowel + number of arrangements with 1 vowel)

Total number of arrangements = 266

Number of arrangements with no vowel = 216 *(6!/6!)

Number of arrangements with 1 vowel = 5*215 * (6!/5!)

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Considering Vowels and Consonants can be repeated, then total 6 letter strings that can be formed :-

$$C(6,2) * 5 * 5 * C(4,4) * 21 * 21 * 21 * 21$$
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There are 6 locations and out of these 6 locations two vowels can be chosen in 6C2 ways.

In the two locations of the vowels, vowels can be chosen in 5*5 ways.

now remaining 4 locations can be filled with 21*21*21*21 ways.

So, no of strings of six lowercase letters of the English alphabet that  contain exactly two vowels =6C2 * 5*5 * 21*21*21*21 =72930375
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$b)$ Say, it can be a repeated characters string or non repeated character string or combination of repeated and non repeated character string

Firstly when always vowels are repeating

Case $1$: Say all vowel are repeating and all non vowel are repeating ${\color{Blue}{----}}{\color{Orange} {--}}$=$\binom{5}{1}\binom{21}{1}\frac{6!}{4!2!}=1575$

Case $2$: Say, all vowel are repeating and 1 consonant is repeating 3 times${\color{Blue}{---}}{\color{Cyan} -}{\color{Orange} {--}}$$\binom{5}{1}\binom{21}{1}\binom{20}{1}\frac{6!}{3!2!}$

                                                                                                                    $=5\times 21\times 20\times \frac{6\times 5\times 4}{2}=126000$

Case $3$: Say, 2 consonant are repeating 2 times=${\color{Blue} {--}}{\color{Teal} {--}}{\color{Orange} {--}}$$\binom{5}{1}\times \binom{21}{2}\frac{6!}{2!\times 2!\times 2!}=94500$

Case $4$: Say, no consonant are repeating$\binom{5}{1}\times \binom{21}{4}\frac{6!}{2!}=10,773,000$

Case $5$: Say, if there are three consonant , where $2$ repeating and $2$ non repeating ${\color{Blue} {--}}{\color{Magenta} {-}}{\color{DarkGreen} {-}}{\color{Orange} {--}}$

$\binom{5}{1}\times \binom{21}{1}\times \binom{20}{1}\times \binom{19}{1}\times \frac{6!}{2!\times 2!}=7,182,000$

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All these 5 cases when no vowels are repeating

Case $1$: Say all vowel are non repeating and all non vowel are repeating$\binom{5}{2}\binom{21}{1}\frac{6!}{4!}=6300$

Case $2$: Say, no vowel are repeating and 1 consonant is repeating 3 times=$\binom{5}{2}\binom{21}{1}\binom{20}{1}\frac{6!}{3!}=504000$

Case $3$: Say, $2$ consonant are repeating $2$ times=$\binom{5}{2}\binom{21}{2}\frac{6!}{2!2!}=378000$

Case $4$: Say, no consonant are repeating=$\binom{5}{2}\binom{21}{4}\times 6!=43,092,000$

Case $5$: Say, if there are three consonant , where $2$ repeating and $2$ non repeating=$\binom{5}{2}\times \binom{21}{1}\binom{20}{1}\binom{19}{1}\frac{6!}{2!}=28,728,000$

Now , summation of all these will be 90,883,800

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