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Elena is managing a construction project. She has 7 workers available to work on a given day, and she needs to select 5 of them.
How many different sets of 5 workers can Elena choose?

Elena is managing a construction project. She has $7$ workers available to work on a given day, and she needs to select $5$ of them. $\newline$ How many different sets of $5$ workers can Elena choose?

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Counting principle

Full solution

Q. Elena is managing a construction project. She has

7

workers available to work on a given day, and she needs to select

5

of them.

\newline

How many different sets of

5

workers can Elena choose?

Identify problem type: Identify the type of problem. Elena is choosing $5$ workers out of $7$ without regard to the order in which they are chosen. This is a combination problem, not a permutation, because the order does not matter.
Use combination formula: Use the combination formula to calculate the number of different sets. The number of ways to choose $k$ items from a set of $n$ items is given by the combination formula $nCk = \frac{n!}{k! \times (n - k)!}$ , where $!$ denotes factorial.
Apply combination formula: Apply the combination formula to the given numbers. Elena has $7$ workers and needs to choose $5$ , so we need to calculate $7C5 = \frac{7!}{5! \times (7 - 5)!}$ .
Simplify factorials: Simplify the factorials in the formula. $\newline$ Calculate $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ , $5! = 5 \times 4 \times 3 \times 2 \times 1$ , and $(7 - 5)! = 2! = 2 \times 1$ .
Cancel common terms: Cancel out the common terms in the numerator and the denominator. $7!$ has $5!$ in it, so we can cancel $5!$ from both the numerator and the denominator. This leaves us with $7C5 = (7 \times 6) / (2 \times 1)$ .
Perform calculation: Perform the calculation. Calculate $(7 \times 6) / (2 \times 1) = 42 / 2 = 21$ .
Conclude with final answer: Conclude with the final answer. Elena can choose $21$ different sets of $5$ workers from the $7$ available workers.

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