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 ofter 5 fruits to Anil is______

Answer 1

Correct Answer: 6860

Possible selections are:

(1) 2 oranges, 1 red apple, 2 white apples.

(2) 2 oranges, 2 red apples, 1 white apple.

(3) 3 oranges, 1 red apple, 1 white apple.

The total number of ways for each case:

\( (1) \quad \binom{8}{2} \cdot \binom{7}{1} \cdot \binom{5}{2} = 1960. \)

\( (2) \quad \binom{8}{2} \cdot \binom{7}{2} \cdot \binom{5}{1} = 2940. \)

\( (3) \quad \binom{8}{3} \cdot \binom{7}{1} \cdot \binom{5}{1} = 1960. \)

Adding them:

1960 + 2940 + 1960 = 6860.

Final Answer:

\( \boxed{6860} \)

Answer 2

The correct answer is 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 $(3O,1RA,1WA)$
$\Rightarrow {}^8{C}_2{}^7{C}_1{}^5{C}_2+{}^8{C}_2{}^7{C}_2{}^5{C}_1+{}^8{C}_3{}^7{C}_1{}^5{C}_1$
$\Rightarrow 1960+2940+1960$
$\Rightarrow 6860$