Question

1a. How many arrangements are there of all the letters in INDIVIDUAL?

1b. How many arrangements of the letters in INDIVIDUAL have all three I’s adjacent?

1c. How many arrangements of the letters in INDIVIDUAL have no I’s adjacent?

Answer 1

(1a)

Letter I = 3 times

Letter N = 1 time

Letter D = 2 times

Letter V = 1 time

Letter U = 1 time

Letter A = 1 time

Letter L =1 time

Total letters = 10 Nos

So,

Number of arrangements are there of all the letters in INDIVIDUAL is given by:

So,

Answer is:

302400

(1b)

Since all three I’s are adjacent, consider the three I’s as one letter.

This, we have:

Letter III = 1 time

Letter N = 1 time

Letter D = 2 times

Letter V = 1 time

Letter U = 1 time

Letter A = 1 time

Letter L =1 time

Total letters = 8 Nos

So,

Number of arrangements are there of all the letters in INDIVIDUAL with all three I’s adjacent is given by:

So,

Answer is:

20160

(1c)

The I's have to be placed in the dashes:

- N - D - D - V - U - A - L -

The number of possible arrangements of the remaining letters is:

The dashes can be filled in:

So,

Number of arrangements of the letters in INDIVIDUAL have no I’s adjacent = 2520 X 56 = 141120

So,

Answer is:

141120