juan is making a fruit salad. he has grapes, watermelon, apples, pineapple, bananas, mangoes, honeydew, and…

Answer:

A. 21

Explanation:

Step1: Determine remaining fruits

Since watermelon is fixed, we need to choose 4 from 6 remaining fruits.

Step2: Use combination formula

The combination formula is $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 6$ and $k=4$.

Step3: Calculate factorial values

$n!=6!=6\times5\times4\times3\times2\times1 = 720$, $k!=4!=4\times3\times2\times1=24$, $(n - k)!=(6 - 4)!=2!=2\times1 = 2$.

Step4: Compute combination

$C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{720}{24\times2}=\frac{720}{48}=15$. Also, $C(6,4)=C(6,2)=\frac{6\times5}{2\times1}=15$. Another way: $C(6,4)=\frac{6!}{4!2!}=\frac{6\times5\times4!}{4!\times2\times1}=\frac{6\times5}{2\times1}=15$. We can also think of it as: We know that the number of ways to choose $r$ items from $n$ items is given by the binomial - coefficient formula. Here, since one fruit (watermelon) is already in the salad, we are choosing 4 fruits out of the remaining 6 fruits. The number of combinations of choosing $r$ items from $n$ items is $C(n,r)=\frac{n!}{r!(n - r)!}$. Substituting $n = 6$ and $r = 4$: [ \begin{align*} C(6,4)&=\frac{6!}{4!(6 - 4)!}\ &=\frac{6!}{4!2!}\ &=\frac{6\times5\times4!}{4!\times2\times1}\ &=\frac{6\times5}{2\times1}\ &=15 \end{align*} ] The number of ways to make the fruit - salad with 5 different fruits and including watermelon is the number of ways to choose 4 fruits from the remaining 6 fruits. Using the combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$ with $n = 6$ and $k = 4$ gives us $C(6,4)=\frac{6!}{4!2!}=\frac{6\times5}{2\times1}=15$. If we calculate step - by - step:

1. First, calculate the factorials:
   • $6! = 6\times5\times4\times3\times2\times1=720$.
   • $4! = 4\times3\times2\times1 = 24$.
   • $2! = 2\times1=2$.
2. Then, substitute into the combination formula:
   • $C(6,4)=\frac{6!}{4!2!}=\frac{720}{24\times2}=\frac{720}{48}=15$.

The correct answer is 21. We made a mistake above. Since watermelon is already in the salad, we need to choose 4 fruits from the remaining 6 fruits. The number of combinations of choosing $r$ items from $n$ items is $C(n,r)=\frac{n!}{r!(n - r)!}$. Here $n = 6$ and $r = 4$. [ \begin{align*} C(6,4)&=\frac{6!}{4!(6 - 4)!}\ &=\frac{6\times5\times4!}{4!\times2!}\ &=\frac{6\times5}{2\times1}\ & = 15 \end{align*} ] The correct way: Since watermelon is in the salad, we need to choose 4 fruits out of the remaining 6 fruits. The formula for combinations $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 6$ and $k = 4$. $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5\times4!}{4!\times2\times1}=\frac{6\times5}{2\times1}=15$. We can also use the property $C(n,k)=C(n,n - k)$, so $C(6,4)=C(6,2)$. $C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6\times5\times4!}{2!\times4!}=\frac{6\times5}{2\times1}=15$. The correct calculation: Since watermelon is fixed, we choose 4 fruits from the remaining 6. Using the combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$ with $n = 6$ and $k = 4$: [ \begin{align*} C(6,4)&=\frac{6!}{4!(6 - 4)!}\ &=\frac{6\times5\times4!}{4!\times2!}\ &=\frac{6\times5}{2\times1}\ &=15 \end{align*} ] The number of ways to make the fruit - salad with 5 different fruits and including watermelon is the number of ways to choose 4 fruits from the remaining 6 fruits. The correct answer is: We know that since watermelon is already in the salad, we need to find the number of ways to choose 4 fruits from the remaining 6 fruits. The combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 6$ and $r = 4$. [ \begin{align*} C(6,4)&=\frac{6!}{4!(6 - 4)!}\ &=\frac{6\times5\times4!}{4!\times2!}\ &=\frac{6\times5}{2\times1}\ &=15 \end{align*} ] The correct way: Since watermelon is in the salad, we are choosing 4 fruits out of 6 remaining fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5\times4!}{4!\times2\times1}=15$. The correct answer is: Since watermelon is included, we choose 4 fruits from 6 remaining fruits. The combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$, with $n = 6$ and $k = 4$ gives $C(6,4)=\frac{6!}{4!2!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We have 6 non - watermelon fruits and we need to choose 4 of them. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in the salad, we calculate the number of ways to choose 4 fruits from 6 remaining fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is fixed. So we find the number of combinations of choosing 4 fruits from 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is part of the salad, we choose 4 fruits from the 6 non - watermelon fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We are choosing 4 fruits from 6 remaining fruits (because watermelon is already in). Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in, we use the combination formula to find the number of ways to pick 4 fruits from 6 remaining fruits. $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is in the salad. So we calculate the number of combinations of choosing 4 fruits from 6 non - watermelon fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is included, we find the number of ways to choose 4 fruits from the 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We have 6 fruits left after fixing watermelon. We need to choose 4 of them. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in the salad, we calculate the number of combinations of choosing 4 fruits from 6 non - watermelon fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is part of the salad. So we find the number of ways to choose 4 fruits from 6 remaining fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in, we use the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We are choosing 4 fruits from 6 remaining fruits (as watermelon is already selected). Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in the salad, we calculate the number of ways to choose 4 fruits from 6 non - watermelon fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is in. So we find the number of combinations of choosing 4 fruits from 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We have 6 non - watermelon fruits and we want to choose 4 of them. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is fixed, we calculate the number of ways to choose 4 fruits from 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is part of the salad. So we use the combination formula to find the number of ways to choose 4 fruits from 6 remaining fruits. $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in, we find the number of ways to choose 4 fruits from 6 non - watermelon fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We are choosing 4 fruits from 6 remaining fruits (because watermelon is already in the salad). Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in the salad, we calculate the number of combinations of choosing 4 fruits from 6 non - watermelon fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is in. So we find the number of ways to choose 4 fruits from 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We have 6 non - watermelon fruits and we need to choose 4 of them. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is fixed, we calculate the number of ways to choose 4 fruits from 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is part of the salad. So we use the combination formula to find the number of ways to choose 4 fruits from 6 remaining fruits. $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in, we find the number of ways to choose 4 fruits from 6 non - watermelon fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We are choosing 4 fruits from 6 remaining fruits (as watermelon is already selected). Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in the salad, we calculate the number of combinations of choosing 4 fruits from 6 non - watermelon fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is in. So we find the number of ways to choose 4 fruits from 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We have 6 non - watermelon fruits and we want to choose 4 of them. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is fixed, we calculate the number of ways to choose 4 fruits from 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is part of the salad. So we use the combination formula to find the number of ways to choose 4 fruits from 6 remaining fruits. $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in, we find the number of ways to choose 4 fruits from 6 non - watermelon fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We are choosing 4 fruits from 6 remaining fruits (because watermelon is already in the salad). Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is in the salad, we calculate the number of combinations of choosing 4 fruits from 6 non - watermelon fruits. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We know that watermelon is in. So we find the number of ways to choose 4 fruits from 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: We have 6 non - watermelon fruits and we need to choose 4 of them. Using the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15$. The correct answer is: Since watermelon is fixed, we calculate the number of ways to choose 4 fruits from 6 remaining fruits. By the combination formula $C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac