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There are 26 students in a homeroom. How many different ways can they be chosen to be elected President, Vice President, and Treasurer?
Answer:

There are $26$ 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

26

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 $26$ 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 Formula: Set up the permutation formula. $\newline$ The number of ways to arrange $n$ items into $r$ specific places is given by the permutation formula: $\newline$ $P(n, r) = \frac{n!}{(n - r)!}$ $\newline$ Where $n$ is the total number of items to choose from, $r$ is the number of items to choose, and “ $!$ ” denotes factorial.
Apply Formula: Apply the permutation formula to our problem. $\newline$ In this case, $n = 26$ (the total number of students) and $r = 3$ (the number of positions to fill). $\newline$ $P(26, 3) = \frac{26!}{(26 - 3)!}$
Calculate Permutation: Calculate the permutation. $\newline$ $P(26, 3) = \frac{26!}{23!}$ $\newline$ To avoid calculating the large factorials, we can expand the factorials and cancel out the common terms. $\newline$ $P(26, 3) = \frac{26 \times 25 \times 24 \times 23!}{23!}$ $\newline$ The $23!$ in the numerator and denominator cancel each other out. $\newline$ $P(26, 3) = 26 \times 25 \times 24$
Perform Multiplication: Perform the multiplication. $\newline$ P $26, 3$ = $26 \times 25 \times 24$ $\newline$ = $15600$ $\newline$ So, there are $15,600$ different ways to choose a President, Vice President, and Treasurer from $26$ students.

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