Question

If four dice are thrown together, then the probability that the sum of the numbers appearing on them is 13, is

• A. $\frac { 5 } { 216 }$
• B. $\frac { 11 } { 216 }$
• C. $\frac { 35 } { 324 }$
• D. $\frac { 1 } { 432 }$

Answer 1

Answer: (C)

$\frac { 35 } { 324 }$

Answer 2

Total number of elementary events associated to the random experiment of throwing 4 dice is

$6 \times 6 \times 6 \times 6 = 6 ^ { 4 }$

Favourable number of elementary events.

$=$ Coeff. of $x ^ { 13 }$ in $\left( x + x ^ { 2 } + x ^ { 3 } + \ldots \ldots . . x ^ { 6 } \right) ^ { 4 }$

$=$ Coeff. of $x ^ { 9 }$ in $\left( 1 + x + x ^ { 2 } + \ldots \ldots + x ^ { 5 } \right) ^ { 4 }$

$=$ Coeff. of $x ^ { 9 }$ in $\left( \frac { 1 - x ^ { 6 } } { 1 - x } \right) ^ { 4 }$

$=$ Coeff. of $x ^ { 9 }$ in $\left( 1 - x ^ { 6 } \right) ^ { 4 } ( 1 - x ) ^ { - 4 }$

$=$ Coeff. of $x ^ { 9 }$ in $( 1 - x ) ^ { - 4 } - 4$ coeff. of $x ^ { 3 }$ in $( 1 - x ) ^ { - 4 }$

= 220-80=140.

Hence, required probability $= \frac { 140 } { 6 ^ { 4 } } = \frac { 35 } { 324 }$