Let A_1, A_2, ......, A_{11} be players in a | vedclass

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(D) Let $x_i$ be the number of coins received by player $A_i$ where $i in {1, 2, ....., 11}$.
The total number of coins is $\sum_{i=1}^{11} x_i = 100

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$^{27}C_{10}$

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(D) Let $x_i$ be the number of coins received by player $A_i$ where $i in {1, 2, ....., 11}$.
The total number of coins is $\sum_{i=1}^{11} x_i = 100$.
According to the conditions:
For $i=3, 4, ....., 11$,$x_i \ge i+1$.
For the captain $(A_1)$,$x_1 \ge 1+5 = 6$.
For the vice-captain $(A_2)$,$x_2 \ge 2+3 = 5$.
Let $x_i = (i+1) + y_i$ for $i=3, ....., 11$,$x_1 = 6 + y_1$,and $x_2 = 5 + y_2$,where $y_i \ge 0$.
Substituting these into the sum:
$(6+y_1) + (5+y_2) + \sum_{i=3}^{11} (i+1+y_i) = 100$
$11 + y_1 + y_2 + \sum_{i=3}^{11} (i+1) + \sum_{i=3}^{11} y_i = 100$
The sum $\sum_{i=3}^{11} (i+1) = 4+5+.....+12 = \frac{9}{2}(4+12) = 72$.
So,$11 + 72 + \sum_{i=1}^{11} y_i = 100 \implies 83 + \sum_{i=1}^{11} y_i = 100$.
Thus,$\sum_{i=1}^{11} y_i = 17$.
The number of non-negative integer solutions is given by the stars and bars formula $\binom{n+k-1}{k-1}$ where $n=17$ and $k=11$.
Number of ways $= \binom{17+11-1}{11-1} = \binom{27}{10}$.

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