Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of

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Solution

Q.
Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is _______ . [2023]

Ans.
(6860)

$Let 7 red apples be denoted by (A), 5 white apples by (B), 8 oranges by (C) .$

$Now, 5 fruits are to be selected (Note: fruits taken different).$

$Possible selections: (2 C, 1 A, 2 B) or (2 C, 2 A, 1 B) or (3 C, 1 A, 1 B)$

$= {}^{8}{C_{2}} {}^{7}{C_{1}} {}^{5}{C_{2}} + {}^{8}{C_{2}} {}^{7}{C_{2}} {}^{5}{C_{1}} + {}^{8}{C_{3}} {}^{7}{C_{1}} {}^{5}{C_{1}}$

$= 28 \times 7 \times 10 + 28 \times 21 \times 5 + 56 \times 7 \times 5$

$= 1960 + 2940 + 1960 = 6860$

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