The total number of ways of selecting six coi | vedclass

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Question

(A) The problem asks for the number of ways to select $6$ coins from $3$ different types of coins,where each type has enough coins (at least $6$) to s

Vedclass · Accepted Answer

$28$

Vedclass · Answer

(A) The problem asks for the number of ways to select $6$ coins from $3$ different types of coins,where each type has enough coins (at least $6$) to satisfy the selection.This is a problem of finding the number of non-negative integer solutions to the equation $x_1 + x_2 + x_3 = 6$,where $x_1, x_2, x_3$ represent the number of one rupee,fifty paise,and twenty-five paise coins selected,respectively.Using the stars and bars formula for combinations with repetition,the number of ways is given by $^{n + r - 1}C_r$,where $n = 3$ (types of coins) and $r = 6$ (number of coins to select).Therefore,the number of ways is $^{3 + 6 - 1}C_6 = ^8C_6$.Using the property $^nC_r = ^nC_{n-r}$,we have $^8C_6 = ^8C_2 = \frac{8 	imes 7}{2 	imes 1} = 28$.

The total number of ways of selecting six coins out of $20$ one rupee coins,$10$ fifty paise coins,and $7$ twenty-five paise coins is:

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