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Amara needs to create a special tasting menu at her restaurant. She needs to select 4 dishes from 7 available dishes and put them in a tasty sequence.
How many unique ways are there to arrange 4 of the 7 dishes?

Amara needs to create a special tasting menu at her restaurant. She needs to select $4$ dishes from $7$ available dishes and put them in a tasty sequence. $\newline$ How many unique ways are there to arrange $4$ of the $7$ dishes?

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Counting principle

Full solution

Q. Amara needs to create a special tasting menu at her restaurant. She needs to select

4

dishes from

7

available dishes and put them in a tasty sequence.

\newline

How many unique ways are there to arrange

4

of the

7

dishes?

Problem Understanding: Understand the problem. $\newline$ Amara needs to select $4$ dishes out of $7$ and arrange them in a sequence. This is a permutation problem because the order of the dishes matters.
Permutation Formula: Apply the formula for permutations without repetition. $\newline$ The number of ways to arrange $k$ items from a set of $n$ items is given by the formula $\frac{n!}{(n-k)!}$ where $＂!＂$ denotes factorial. $\newline$ Here, $n = 7$ (total dishes) and $k = 4$ (dishes to be arranged).
Calculate $n!$ : Calculate the factorial of $n$ ( $7!$ ). $\newline$ $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Calculate $(n-k)!$ : Calculate the factorial of $(n-k)$ which is $(7-4)!$ or $3!$ . $3! = 3 \times 2 \times 1$
Substitute into Formula: Substitute the values into the permutation formula. $\newline$ Number of unique ways = $7! / (7-4)!$ $\newline$ = $7! / 3!$
Perform Calculation: Perform the calculation. $\newline$ Number of unique ways $= (7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) / (3 \times 2 \times 1)$ $\newline$ The $3 \times 2 \times 1$ in the denominator and numerator cancel each other out. $\newline$ Number of unique ways $= 7 \times 6 \times 5 \times 4$ $\newline$ $= 840$

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