Question

Math

Posted 5 months ago

```
Manny's burrito stand lets customers choose 4 ingredients from the list below to add their burritos:
\begin{tabular}{|c|c|c|c|}
\hline Onions & Cilantro & Jalapeño & Beans \\
\hline Cheese & Crema & Salsa & Hot sauce \\
\hline
\end{tabular}
How many groups of 4 different ingredients can go on a burrito?
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

To determine the number of different groups of 4 ingredients that can be chosen from 8 ingredients, we use the binomial coefficient

step 2

The binomial coefficient formula is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of items, and $k$ is the number of items to choose

step 3

Applying the formula with $n=8$ and $k=4$, we get $\binom{8}{4} = \frac{8!}{4!(8-4)!}$

step 4

Simplifying the expression, we have $\binom{8}{4} = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1}$

step 5

Calculating the simplified expression gives us $\binom{8}{4} = 70$

Answer

There are 70 different groups of 4 ingredients that can go on a burrito.

Key Concept

Combinations and the Binomial Coefficient

Explanation

The number of ways to choose $k$ items from a set of $n$ distinct items is given by the binomial coefficient $\binom{n}{k}$, which is calculated using the formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. This represents the number of combinations possible without regard to order.

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