How many different ways can you choose 4 fruits from a basket of 10 fruits if you don't want to choose any of one type of fruit?

Sure, let's break it down step by step.

1. Identify the total number of fruits and the restriction:

   • There are 10 fruits in total.
   • We need to choose 4 fruits.
   • We don't want to choose any of one type of fruit.

2. Calculate the total number of ways to choose 4 fruits from 10 without any restrictions:

   • This can be done using the combination formula $\binom{n}{k}$, where $n$ is the total number of items, and $k$ is the number of items to choose.
   • Here, $n = 10$ and $k = 4$.
   • So, $\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!}$.

3. Calculate $\binom{10}{4}$:

   • $10! = 10 \times 9 \times 8 \times 7 \times 6!$.
   • $4! = 4 \times 3 \times 2 \times 1 = 24$.
   • $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$.
   • So, $\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \times 6!} = \frac{10 \times 9 \times 8 \times 7}{24}$.

4. Simplify the fraction:

   • $10 \times 9 \times 8 \times 7 = 5040$.
   • $\frac{5040}{24} = 210$.

5. Account for the restriction:

   • We need to subtract the number of ways to choose 4 fruits that include at least one of the restricted type.
   • Assume there is 1 type of fruit we don't want to choose, and there are 9 other fruits left.
   • We need to choose 4 fruits from these 9 fruits.
   • So, we calculate $\binom{9}{4}$.

6. Calculate $\binom{9}{4}$:

   • $\binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4! \cdot 5!}$.
   • $9! = 9 \times 8 \times 7 \times 6 \times 5!$.
   • $4! = 24$.
   • $5! = 120$.
   • So, $\binom{9}{4} = \frac{9 \times 8 \times 7 \times 6 \times 5!}{4! \times 5!} = \frac{9 \times 8 \times 7 \times 6}{24}$.

7. Simplify the fraction:

   • $9 \times 8 \times 7 \times 6 = 3024$.
   • $\frac{3024}{24} = 126$.

8. Subtract the restricted combinations from the total combinations:

   • Total combinations without restriction: 210.
   • Combinations including the restricted type: 126.
   • So, the number of ways to choose 4 fruits without choosing any of one type: $210 - 126 = 84$.

Therefore, there are 84 different ways to choose 4 fruits from a basket of 10 fruits if you don't want to choose any of one type of fruit.