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 orange, 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 _____.

We need to find the number of ways to select 5 fruits from 7 red apples, 5 white apples, and 8 oranges, with at least 2 oranges, at least 1 red apple, and at least 1 white apple.

Let $$r$$, $$w$$, and $$o$$ be the number of red apples, white apples, and oranges respectively. We need:

• $$r + w + o = 5$$
• $$o \geq 2$$, $$r \geq 1$$, $$w \geq 1$$
• $$r \leq 7$$, $$w \leq 5$$, $$o \leq 8$$

The possible cases are:

Case 1: $$o = 2$$, $$r + w = 3$$ with $$r \geq 1, w \geq 1$$.

Possible: $$(r, w) \in \{(1, 2), (2, 1)\}$$

• $$(1, 2, 2)$$: $$\binom{7}{1} \cdot \binom{5}{2} \cdot \binom{8}{2} = 7 \times 10 \times 28 = 1960$$
• $$(2, 1, 2)$$: $$\binom{7}{2} \cdot \binom{5}{1} \cdot \binom{8}{2} = 21 \times 5 \times 28 = 2940$$

Case 2: $$o = 3$$, $$r + w = 2$$ with $$r \geq 1, w \geq 1$$.

Only $$(r, w) = (1, 1)$$:

• $$(1, 1, 3)$$: $$\binom{7}{1} \cdot \binom{5}{1} \cdot \binom{8}{3} = 7 \times 5 \times 56 = 1960$$

$$1960 + 2940 + 1960 = 6860$$

The correct answer is $$6860$$.