Key Concepts and Formulas

• Combinations: The number of ways to choose $r$ objects from a set of $n$ distinct objects is given by the binomial coefficient: ${n \choose r} = \frac{n!}{r!(n-r)!}$
• Addition Principle: If there are $m$ ways to do one thing and $n$ ways to do another, and these things are mutually exclusive, then there are $m+n$ ways to do either one.

Step-by-Step Solution

Step 1: Identify the constraints

Anil's mother needs to select 5 fruits such that there are at least 2 oranges, at least 1 red apple, and at least 1 white apple.

Step 2: Enumerate all possible cases based on the number of oranges

Let $o$ be the number of oranges, $r$ be the number of red apples, and $w$ be the number of white apples. We need to find all possible combinations of $o, r, w$ such that $o+r+w = 5$, $o \ge 2$, $r \ge 1$, and $w \ge 1$.

• Case 1: $o=2$. Then $r+w = 3$. Since $r \ge 1$ and $w \ge 1$, we have two subcases:
  • $r=1, w=2$
  • $r=2, w=1$
• Case 2: $o=3$. Then $r+w = 2$. Since $r \ge 1$ and $w \ge 1$, we have only one subcase:
  • $r=1, w=1$

Therefore, the possible distributions are: * 2 oranges, 1 red apple, 2 white apples * 2 oranges, 2 red apples, 1 white apple * 3 oranges, 1 red apple, 1 white apple

Step 3: Calculate the number of ways for each case

• Case 1: 2 oranges, 1 red apple, 2 white apples The number of ways to choose 2 oranges from 8 is ${8 \choose 2}$. The number of ways to choose 1 red apple from 7 is ${7 \choose 1}$. The number of ways to choose 2 white apples from 5 is ${5 \choose 2}$. Therefore, the total number of ways for this case is: ${8 \choose 2} \times {7 \choose 1} \times {5 \choose 2} = \frac{8!}{2!6!} \times \frac{7!}{1!6!} \times \frac{5!}{2!3!} = 28 \times 7 \times 10 = 1960$

• Case 2: 2 oranges, 2 red apples, 1 white apple The number of ways to choose 2 oranges from 8 is ${8 \choose 2}$. The number of ways to choose 2 red apples from 7 is ${7 \choose 2}$. The number of ways to choose 1 white apple from 5 is ${5 \choose 1}$. Therefore, the total number of ways for this case is: ${8 \choose 2} \times {7 \choose 2} \times {5 \choose 1} = \frac{8!}{2!6!} \times \frac{7!}{2!5!} \times \frac{5!}{1!4!} = 28 \times 21 \times 5 = 2940$

• Case 3: 3 oranges, 1 red apple, 1 white apple The number of ways to choose 3 oranges from 8 is ${8 \choose 3}$. The number of ways to choose 1 red apple from 7 is ${7 \choose 1}$. The number of ways to choose 1 white apple from 5 is ${5 \choose 1}$. Therefore, the total number of ways for this case is: ${8 \choose 3} \times {7 \choose 1} \times {5 \choose 1} = \frac{8!}{3!5!} \times \frac{7!}{1!6!} \times \frac{5!}{1!4!} = 56 \times 7 \times 5 = 1960$

Step 4: Calculate the total number of ways

Since these cases are mutually exclusive, we can add the number of ways for each case to get the total number of ways. $1960 + 2940 + 1960 = 6860$

The provided answer is incorrect. Let's re-examine the cases and ensure no overcounting or undercounting. The cases are correctly identified. The calculations are correct. However, the provided "Correct Answer" is 8, which is significantly different from our calculated answer.

Let's revisit the problem statement and the constraints. $o+r+w = 5$ $o \ge 2, r \ge 1, w \ge 1$

The possible combinations are: (2, 1, 2): 2 oranges, 1 red apple, 2 white apples. Number of ways = ${8 \choose 2} {7 \choose 1} {5 \choose 2} = 28 \times 7 \times 10 = 1960$ (2, 2, 1): 2 oranges, 2 red apples, 1 white apple. Number of ways = ${8 \choose 2} {7 \choose 2} {5 \choose 1} = 28 \times 21 \times 5 = 2940$ (3, 1, 1): 3 oranges, 1 red apple, 1 white apple. Number of ways = ${8 \choose 3} {7 \choose 1} {5 \choose 1} = 56 \times 7 \times 5 = 1960$

Total number of ways = $1960 + 2940 + 1960 = 6860$

Since we are certain about our cases and calculations, and the "Correct Answer" given is 8, it is highly likely that the correct answer provided is wrong.

Let us try to reach 8. This seems impossible with the given constraints.

Common Mistakes & Tips

• Make sure to identify all possible cases correctly. A good way is to fix one variable (e.g., the number of oranges) and then find the possible values for the other variables.
• Double-check your calculations to avoid arithmetic errors.
• Ensure that the cases are mutually exclusive, so you can simply add the number of ways for each case.

Summary

We identified all possible distributions of oranges, red apples, and white apples such that the given constraints are satisfied. We then calculated the number of ways for each case and added them together to get the total number of ways. Our calculated answer is 6860. Since the provided answer is 8, which is very unlikely to be correct, we suspect that the provided answer is incorrect.

Final Answer

The problem is likely flawed, as the calculated number of ways, 6860, does not match the provided answer of 8. The final answer is \boxed{6860}. The provided "Correct Answer" is incorrect.