The number of ways to roll 5 six sided dice to get sum of 25 is________
126

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Solution

The correct option is A 126
Number of possible values on top of dice:
$= x + x^{2} + x^{3} + x^{4} + x^{5} + x^{6}$
$= \frac{x (1 - x^{6})}{1 - x}$

$W e n e e d t o f i n d c o e f f i c i e n t o f x^{25} :$
${(\frac{x (1 - x^{6})}{1 - x})}^{5} = x^{5} (1 - x^{6})^{5} . \frac{1}{(1 - x)^{5}} = x^{25}$

$\Rightarrow C o e f f i c i e n t o f x^{20} i n (1 - x^{6})^{5} \cdot \frac{1}{(1 - x)^{5}}$

$\Rightarrow C o e f f i c i e n t o f x^{20} i n (1 - 5 x^{6} + 10 x^{12} - 10 x^{18} + 5 x^{24} - x^{30}) (ω \sum n = 0 [\begin{matrix} n + 4 4 \end{matrix}] x^{n})$

$\Rightarrow C o e f f i c i e n t o f x^{20} i n {(^{20 + 4} C_{4}) - 5 \times (^{14 - 4} C_{4}) + 10 \times (^{8 - 4} C_{4}) - 10 \times (^{2 - 4} C_{4})] x^{20}$
$\Rightarrow = (^{24} C_{4}) - 5 \times (^{18} C_{4}) + 10 \times (^{12} C_{4}) - 10 \times (^{6} C_{4})$
$= 10626 - 5 \times (3060) + 10 \times (495) - 10 \times (15)$
= 126

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