JEE MAIN - Mathematics (2023 - 25th January Evening Shift - No. 22)
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 ____________
Answer
6860 OR 3
Explanation
Total 8 oranges, 5 white apple and 7 red apple. 5 fruits needs to be selected.
Case I: 3 orange $+1$ red apple $+1$ white apple
$$ ={ }^{8} C_{3} \times{ }^{7} C_{1} \times{ }^{5} C_{1}=1960 $$
Case II : 2 oranges $+2$ red apples $+1$ white apple.
$$ ={ }^{8} C_{2} \times{ }^{7} C_{2} \times{ }^{5} C_{1}=2940 $$
Case III : 2 oranges $+1$ red apples $+2$ white apple.
$$ \begin{aligned} & ={ }^{8} C_{2} \times{ }^{7} C_{1} \times{ }^{5} C_{2} \\\\ & =1960 \\\\ \text { Total } & =1960+2940+1960 \\\\ & =6860 \end{aligned} $$
Case I: 3 orange $+1$ red apple $+1$ white apple
$$ ={ }^{8} C_{3} \times{ }^{7} C_{1} \times{ }^{5} C_{1}=1960 $$
Case II : 2 oranges $+2$ red apples $+1$ white apple.
$$ ={ }^{8} C_{2} \times{ }^{7} C_{2} \times{ }^{5} C_{1}=2940 $$
Case III : 2 oranges $+1$ red apples $+2$ white apple.
$$ \begin{aligned} & ={ }^{8} C_{2} \times{ }^{7} C_{1} \times{ }^{5} C_{2} \\\\ & =1960 \\\\ \text { Total } & =1960+2940+1960 \\\\ & =6860 \end{aligned} $$
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