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6.Permutation and Combination
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Let $A_1,A_2,........A_{11}$ are players in a team with their T-shirts numbered $1,2,.....11$. Hundred gold coins were won by the team in the final match of the series. These coins is to be distributed among the players such that each player gets atleast one coin more than the number on his T-shirt but captain and vice captain get atleast $5$ and $3$ coins respectively more than the number on their respective T-shirts, then in how many different ways these coins can be distributed ?
A
$^{100}{C_{83}}$
B
$^{28}{C_{11}}$
C
$^{27}{C_{9}}$
D
$^{27}{C_{10}}$
Solution
Satisfying initial conditions gold coins remaining $=100-(2+3+4+\ldots+12)-4-2$
$=17$
Now distribution of $17$ coins among $11$ players
$ = {\,^{17 + 11 – 1}}{{\rm{C}}_{11 – 1}}{ = ^{27}}{{\rm{C}}_{10}}$
Standard 11
Mathematics
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