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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 1616 students in a homeroom. How many different ways can they be chosen to be elected President, Vice President, and Treasurer?\newlineAnswer:

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Full solution

Q. There are 1616 students in a homeroom. How many different ways can they be chosen to be elected President, Vice President, and Treasurer?\newlineAnswer:
  1. Understand the Problem: Understand the problem.\newlineWe need to find the number of different ways to choose 33 students from a group of 1616 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 AA for President and student BB for Vice President is different from choosing student BB for President and student AA for Vice President).
  2. Set up the Formula: Set up the permutation formula.\newlineThe number of ways to arrange nn items in rr specific places is given by the permutation formula, which is nPr=n!(n−r)!nPr = \frac{n!}{(n-r)!} where ＂!＂＂!＂ denotes factorial.
  3. Apply the Formula: Apply the permutation formula.\newlineIn this case, n=16n = 16 (the total number of students) and r=3r = 3 (the number of positions to fill). So we need to calculate 16P316P3.\newline16P3=16!(16−3)!16P3 = \frac{16!}{(16-3)!}
  4. Calculate Factorial Difference: Calculate the factorial difference. 16!/(16−3)!16! / (16-3)! simplifies to 16!/13!16! / 13! because (16−3)=13(16-3) = 13.
  5. Simplify the Expression: Simplify the expression.\newlineWe can simplify 16!/13!16! / 13! by canceling out the common factorial terms.\newline16!/13!=(16×15×14×13!)/13!16! / 13! = (16 \times 15 \times 14 \times 13!) / 13!\newlineThe 13!13! in the numerator and denominator cancel out, leaving us with 16×15×1416 \times 15 \times 14.
  6. Perform the Multiplication: Perform the multiplication.\newlineNow we multiply the remaining numbers.\newline16×15×14=240×14=336016 \times 15 \times 14 = 240 \times 14 = 3360
  7. Conclude with Answer: Conclude with the final answer.\newlineThere are 33603360 different ways for the 1616 students to be chosen for the positions of President, Vice President, and Treasurer.

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