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 …………

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answer is 6860.

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Detailed Solution

case	7 red applies	5 white apples	8 oranges	Selection of 5 fruits
I	1	1	3	$^{7} C_{1} \cdot^{5} C_{1} \cdot^{8} C_{3}$
II	1	2	2	$^{7} C_{1} \cdot \cdot^{5} C_{2} \cdot^{8} C_{2}$
III	2	1	2	$^{7} C_{2} \cdot^{5} C_{1} \cdot^{8} C_{2}$

Total number of ways =
$\begin{array}{l} =^{7} C_{1} \cdot^{5} C_{1} \cdot^{8} C_{3} +^{7} C_{1} \cdot^{5} C_{2} \cdot^{8} C_{2} +^{7} C_{2} \cdot^{5} C_{1} \cdot^{8} C_{2} \\ = 7.5 \cdot 56 + 7 \cdot 10.28 + 21.5 \cdot 28 \\ = 7.5 \cdot 28 (2 + 2 + 3) \\ = 7 \times 5 \times 28 \times 7 = 6860 \end{array}$

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