Question

In how many ways the letters of the word PERMUTATIONS can be arranged so that 1). Words start with P and ends with S. 2).vowels in the word are together.

Answer 1

Answer:

1) 20 160

2)  2 419 200

Step-by-step explanation:

Number of letters in PERMUTATIONS = 12

Number of repeats = 2 (There are 2 Ts)

Question 1:

Words start with P and ends with S

Number of letters for permutation = 10 - 2 = 8

Find the number of permutation:

Number of ways = 8!

Number of ways = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320

But there are 2Ts:

Number of ways = 40320 ÷ 2! =  40320/2 = 20 160

Answer: There are 20 160 ways

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Question 2:

Number of vowels = 5 (They are E, U A, I, O)

Number of consonant  = 12 - 5 = 7

Find the number for permutation of the vowels:

Number of ways = 5!

Number of ways = 5 x 4 x 3 x 2 x 1 = 120

Find the number of ways so that the vowels are together

Number of ways = (7 + 1)! x 120

Number of ways = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 120 = 4 838 400

But there are 2 Ts:

Number of ways = 4 838 400 ÷ 2!

Number of ways = 4 838 400/2 = 2 419 200

Answer: There are 2 419 200 ways