In how many ways can the letters of the word PERMUTATIONS be arranged if the (i) Words start with P and end with S, (ii) Vowels are all together
In how many ways can the letters of the word PERMUTATIONS be arranged if the (i) Words start with P and end with S, (ii) Vowels are all together

(i) Total number of letters in PERMUTATIONS

    \[=\text{ }12\]

Only repeated letter is T; 2times

First and last letter of the word are fixed as P and S respectively.

Number of letters remaining

    \[=12\text{ }-\text{ }2\text{ }=\text{ }10\]

⇒ Number of permutations =
NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations Image 23

(ii) Number of vowels in PERMUTATIONS

    \[=\text{ }5\]

(E, U, A, I, O)

Now, we consider all the vowels together as one.

Number of permutations of vowels

    \[=\text{ }120\]

Now total number of letters

    \[=\text{ }12\text{ }-\text{ }5\text{ }+\text{ }1=\text{ }8\]

⇒ Number of permutations =
NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations Image 24

Therefore, total number of permutations

    \[=\text{ }120\text{ }\times \text{ }20160\text{ }=\text{ }2419200\]