A bag contains 23 balls, of which 7 are ident | vedclass

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(B) Let the number of identical balls be $7$ and the number of distinct balls be $23 - 7 = 16$.To select $12$ balls, we can choose $k$ identical balls

Vedclass · Accepted Answer

$^{16}C_5 + ^{16}C_6 + ^{16}C_7 + ^{16}C_8 + ^{16}C_9 + ^{16}C_{10} + ^{16}C_{11} + ^{16}C_{12}$

Vedclass · Answer

(B) Let the number of identical balls be $7$ and the number of distinct balls be $23 - 7 = 16$.To select $12$ balls, we can choose $k$ identical balls where $0 \le k \le 7$.If we choose $k$ identical balls, we must choose $12 - k$ distinct balls from the $16$ distinct ones.The number of ways is $\sum_{k=0}^{7} {^{16}C_{12-k}}$.Expanding this, we get:${^{16}C_{12}} + {^{16}C_{11}} + {^{16}C_{10}} + {^{16}C_{9}} + {^{16}C_{8}} + {^{16}C_{7}} + {^{16}C_{6}} + {^{16}C_{5}}$Rearranging the terms, we get:$^{16}C_5 + ^{16}C_6 + ^{16}C_7 + ^{16}C_8 + ^{16}C_9 + ^{16}C_{10} + ^{16}C_{11} + ^{16}C_{12}$This matches option $(B)$.

$A$ bag contains $23$ balls, of which $7$ are identical. The number of ways of selecting $12$ balls from the bag is:

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