Question

Number of ways of selection of 8 umbrella from 24 umbrellas of which 8 are green color, 8 are blue color and the rest unlike, is given by

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

The number of selections of umbrellas = coefficient of $x^8$ in

$(1+x+x^2+.............+x^8)(1+x+x^2+.............+x^8).(1+x{)}^8$ = coefficient of $x^8$ in $\frac{(1-x^9{)}^2}{(1-x{)}^2}(1+x{)}^8$

= coefficient of $x^8$ in $(1+x{)}^8(1-x{)}^{-2}$

= coefficient of $x^8$ in (${}^8{C}_0{+}^8{C}_{1}x{+}^8{C}_2{x}^2+..............{+}^8{C}_8{x}^8)$

× $(1+2x+3x^2+4x^3+...............+9x^8+..............)$

= ${9.}^8{C}_0+{8.}^8{C}_1+{7.}^8{C}_2+.........+{1.}^8{C}_8$

= ${}^8{C}_0+2^8{C}_1+3^8{C}_2+.............+9^8{C}_8$

Now ${}^8{C}_{0}x{+}^8{C}_1{x}^2+.............{+}^8{C}_8{x}^9=x(1+x{)}^8$

Differentiating with respect to x, we get

${}^8{C}_0+2^8{C}_{1}x+3^8{C}_2{x}^2+..........{.9}^8{C}_8{x}^8=(1+x{)}^8+8x(1+x{)}^7$

Putting x = 1, we get $C_0+2C_1+3C_2+............+9C_8$

= $2^8+{8.2}^7=2^7.(2+8)={10.2}^7$.