Question

In how many ways the sum of upper faces of four distinct dies can be six.

• A. 10
• B.
• C.
• D. 20

Answer 1

The correct option is A

10

Number of ways should be equal to number of solution of $x_1+x_2+x_3+x_4$ = 6

Conditions are $1\le x_1,x_2,x_3,x_4\le 6$

Since the upper limit is 6 which are equal to the sum required, so upper limit can be taken as infinite.

So, number of solution = coefficient of $x^{6}in(x+x^2+x^3+..........{)}^4$

=coefficient of $x^{6}in{x}^4(1-x{)}^{-4}$

= coefficient of $x^{6}in{x}^4$ (1 + ${}^4{C}_1$ x + ${}^5{C}_2$ $x^2$+ ${}^6{C}_3$ $x^3$ + ..............)

= ${}^5{C}_2$ = 10