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There are 16 students in a homeroom. How many different ways can they be chosen to be elected President, Vice President, and Treasurer?
Answer:

There are $16$ students in a homeroom. How many different ways can they be chosen to be elected President, Vice President, and Treasurer? $\newline$ Answer:

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Experimental probability

Full solution

Q. There are

16

students in a homeroom. How many different ways can they be chosen to be elected President, Vice President, and Treasurer?

\newline

Answer:

Understand the Problem: Understand the problem. $\newline$ We need to find the number of different ways to choose $3$ students from a group of $16$ for the positions of President, Vice President, and Treasurer. This is a permutation problem because the order in which we choose the students matters (choosing student $A$ for President and student $B$ for Vice President is different from choosing student $B$ for President and student $A$ for Vice President).
Set up the Formula: Set up the permutation formula. $\newline$ The number of ways to arrange $n$ items in $r$ specific places is given by the permutation formula, which is $nPr = \frac{n!}{(n-r)!}$ where $＂!＂$ denotes factorial.
Apply the Formula: Apply the permutation formula. $\newline$ In this case, $n = 16$ (the total number of students) and $r = 3$ (the number of positions to fill). So we need to calculate $16P3$ . $\newline$ $16P3 = \frac{16!}{(16-3)!}$
Calculate Factorial Difference: Calculate the factorial difference. $16! / (16-3)!$ simplifies to $16! / 13!$ because $(16-3) = 13$ .
Simplify the Expression: Simplify the expression. $\newline$ We can simplify $16! / 13!$ by canceling out the common factorial terms. $\newline$ $16! / 13! = (16 \times 15 \times 14 \times 13!) / 13!$ $\newline$ The $13!$ in the numerator and denominator cancel out, leaving us with $16 \times 15 \times 14$ .
Perform the Multiplication: Perform the multiplication. $\newline$ Now we multiply the remaining numbers. $\newline$ $16 \times 15 \times 14 = 240 \times 14 = 3360$
Conclude with Answer: Conclude with the final answer. $\newline$ There are $3360$ different ways for the $16$ students to be chosen for the positions of President, Vice President, and Treasurer.

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