Question

The number of ways to arrange the letters of the English alphabet, so that there are exactly 5 letters between a and b, is:
1. ²⁴P₅
2. ²⁴P₅ × 20!
3. 2 × ²⁴P₅ × 20!
4. 2 × 24P5 × 24!

Answer 1

Correct Answer - Option 3 : 2 × ²⁴P₅ × 20!

Concept:

Combinations: The number of ways in which r distinct objects can be selected simultaneously from a group of n distinct objects, is:

nCr = \(\rm \frac {n!}{r!(n-r)!}\).

Permutations: The number of ways in which r objects can be arranged in n places (without repetition) is:

ⁿP_r = \(\rm \frac{n!}{(n - r)!}\).

• nPr = nCr × r!.

• n! = 1 × 2 × 3 × ... × n.

• 0! = 1.

 

Calculation:

There are 26 letters in the English alphabet. If we separate the group (a, some 5 letters, b), we will be left with 19 more letters.

These 20 objects (1 group + 19 letters) can be arranged among themselves in 20! ways.

Since either a or b can be at the beginning or the end of the group of 7 letters (a, some 5 letters, b), the number of possible arrangements of the group  will be 2 × (¹P₁ × ⁵P₅ × 1P1) = 2 × 5!.

Also, each group of 5 letters can be selected from the remaining 24 letters (except a and b) in ²⁴C₅ ways.

Required total number of ways = (2 × 5! × 24C5) × 20!

= 2 × 24P5 × 20!.