Question

The number of selections of four letters from the letters of the word ASSASSINATION is-

• A. 72
• B. 71
• C. 66
• D. 52

Answer 1

Answer: (A)

72

Answer 2

The word ASSASSINATION consists of 3 A’s, 4 S’s, 2 I’s and 2N’s, T and O.

The number of selections of four letters

= coefficient of x⁴ in

$\left. \underset { \text { for } 4 S ^ { \prime } s } { \downarrow } + \mathrm { x } ^ { 2 } + \mathrm { x } ^ { 3 } + \mathrm { x } ^ { 4 } \right)$ $\left( 1 + x + x ^ { 2 } \right) ^ { 2 }$ $\underset { \text { for 2I's } } { \downarrow }$ $\left( 1 + x ^ { 2 } \right) ^ { 2 }$ $\downarrow$ for T and O

=Coefficient of x⁴ in $\left\lbrack \frac{1 - x^{4}}{1 - x} \right\rbrack$ $\left\lbrack \frac{1 - x^{5}}{1 - x} \right\rbrack$ $\left\lbrack \frac{1 - x^{3}}{1 - x} \right\rbrack^{2}$ $\left\lbrack \frac{1 - x^{2}}{1 - x} \right\rbrack^{2}$

= Coefficient of x⁴ in

(1 – x⁴) (1 – x⁵) (1– 2x³ + x⁶) (1– 2x⁴ + x⁴) (1– x)^–6

= Coefficient of x⁴ in (1 –2x² – 2x³ + 3x⁵ +….)

(1+ ⁶C₁x + ⁷C₂x² + ⁸C₂x³ + ⁹C₄x⁴ +…..)

= ⁹C₄ –2.⁷C₂ – 2. ⁶C₁ = 126 – 42 – 12 = 72