Question

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 1

Answer: 6860

$7$ Red apple(RA),5 white apple(WA), 8 oranges $(O)$
5 fruits to be selected (Note:- fruits taken different)
Possible selections :- $(2O,1RA,2WA)$ or $(2O$,
$2RA,1WA)$ or $(3O,1RA,1WA)$
$\Rightarrow {}^8{C}_2{}^7{C}_1{}^5{C}_2+{}^8{C}_2{}^7{C}_2{}^5{C}_1+{}^8{C}_3{}^7{C}_1{}^5{C}_1$
$\Rightarrow 1960+2940+1960$
$\Rightarrow 6860$