# 5a) find the number of 5-letter words(with or without any meaning)composed of 3 different consonants and 2 vowels? 5b) ? 5c) ? 5d)? 5e)? ### 2 respuestas

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• The general way to solve all of these is to first count the ways to get the desired set of letters and multiply by the ways to arrange them.

For example, from the 21 consonants, choose 3.

C(21,3) = 21 * 20 * 19 / (3 * 2 * 1)

= 1330 ways

Then from the 5 vowels, choose 2:

C(5,2) = 5 * 4 / (2 * 1) = 10 ways

The total ways to pick a set of 5 letters is the product:

C(21,3) * C(5,2)

= 13,300 ways

Now arrange the 5 letters:

5! = 5 * 4 * 3 * 2 * 1

= 120

Total "words" = 13300 * 120

= 1,596,000

5b) In choosing the letters, you must choose a B --> C(1,1) = 1 way

Then from the remaining *20* consonants, pick 2 more --> C(20,2) = 190 ways

Pick the 2 vowels, then arrange them 5! ways.

5c) C(2,2) = 1 way to pick B and C.

C(19,1) = 19 ways to pick the other consonant.

C(5,2) = 10 ways to pick the vowels.

5! ways to arrange the set of letters.

5d) Take the cases from 5a) and subtract cases where you don't pick B and C. That will leave you cases where at least one is picked. Then arrange them 5! ways.

[C(21,3) - C(19,3)] * C(5,2) * 5!

5e) Count the letters like 5c, but then you fix B and C at the beginning and end (1 way) and then arrange the *3* remaining letters (3! = 6)

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