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 (integer)
6860
Solution
Total 8 oranges, 5 white apple and 7 red apple. 5 fruits needs to be selected.
<br/><br/>
<b>Case I</b>: 3 orange $+1$ red apple $+1$ white apple
<br/><br/>
$={ }^{8} C_{3} \times{ }^{7} C_{1} \times{ }^{5} C_{1}=1960$
<br/><br/>
<b>Case II</b> : 2 oranges $+2$ red apples $+1$ white apple.
<br/><br/>
$={ }^{8} C_{2} \times{ }^{7} C_{2} \times{ }^{5} C_{1}=2940$
<br/><br/>
<b>Case III</b> : 2 oranges $+1$ red apples $+2$ white apple.
<br/><br/>
$$
\begin{aligned}
& ={ }^{8} C_{2} \times{ }^{7} C_{1} \times{ }^{5} C_{2} \\\\
& =1960 \\\\
\text { Total } & =1960+2940+1960 \\\\
& =6860
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Permutations and Combinations · Topic: Fundamental Counting Principle
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