(i) INDEPENDENCE

Here's 12 letters in the word ‘INDEPENDENCE’ out of which 2 are D’s, 3 are N’s, 4 are E’s and the rest all are distinct.

Therefore by using the formula,

n!/ (p! × q! × r!)

The total number of arrangements = 12! / (2! 3! 4!)

= [12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1] / (2! 3! 4!)

= [12 × 11 × 10 × 9 × 8 × 7 × 6 × 5] / (2 × 1 × 3 × 2 × 1)

= 11 × 10 × 9 × 8 × 7 × 6 × 5

= 1663200

(ii) INTERMEDIATE

Here's 12 letters in the word ‘INTERMEDIATE’ out of which 2 are I’s, 2 are T’s, 3 are E’s and the rest all are distinct.

Therefore by using the formula,

n!/ (p! × q! × r!)

The total number of arrangements = 12! / (2! 2! 3!)

= [12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1] / (2! 2! 3!)

= [12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 3 × 2 × 1] / (3!)

= 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5

= 19958400

(iii) ARRANGE

There are 7 letters in the word ‘ARRANGE’ out of which 2 are A’s, 2 are R’s and the rest all are distinct.

Therefore by using the formula,

n!/ (p! × q! × r!)

The total number of arrangements = 7! / (2! 2!)

= [7 × 6 × 5 × 4 × 3 × 2 × 1] / (2! 2!)

= 7 × 6 × 5 × 3 × 2 × 1

= 1260

(iv) INDIA

Here's 5 letters in the word ‘INDIA’ out of which 2 are I’s and the rest all are distinct.

Therefore by using the formula,

n!/ (p! × q! × r!)

The total number of arrangements = 5! / (2!)

= [5 × 4 × 3 × 2 × 1] / 2!

= 5 × 4 × 3

= 60

(v) PAKISTAN

Here's 8 letters in the word ‘PAKISTAN’ out of which 2 are A’s and the rest all are distinct.

Therefore by using the formula,

n!/ (p! × q! × r!)

The total number of arrangements = 8! / (2!)

= [8 × 7 × 6 × 5 × 4 × 3 × 2 × 1] / 2!

= 8 × 7 × 6 × 5 × 4 × 3

= 20160