The number of ways in which a score of $11$ can be made by three persons, each throwing a single dice once, is

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Solution

The correct option is B 27
Required number of ways
$=$ coeffiecient of $x^{11}$ in $(x + x^{2} + . . . + x^{6})^{3}$
$=$ coeffiecient of $x^{8}$ in $(1 + x + . . . + x^{5})^{3}$
$=$ coeffiecient of $x^{8}$ in $(\frac{(1 - x^{6})}{(1 - x)})^{3}$
$=$ coeffiecient of $x^{8}$ in $(1 - x^{18} - 3 x^{6} + 3 x^{12}) (1 + 3 x +^{4} C_{2} x^{2} +^{5} C_{3} x^{3} + . . . . . . +^{7} C_{5} x^{5} + . . . . +^{10} C_{8} x^{8} + . . . .)$
$=^{10} C_{2} - 3 \times^{4} C_{2}$
$= 27$

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