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Difficulty: MediumBasic Probability and Counting Methods

A high school photography club has 1212 active members, consisting of 77 seniors and 55 juniors. The club needs to elect an executive board composed of a President, a Vice President, and a Secretary, where no member can hold more than one position. If the President must be a senior, how many different executive board arrangements are possible?

Answer: 770

Answer

The total number of different executive board arrangements possible is 770770.
To calculate the total number of distinct executive board arrangements, use the Fundamental Counting Principle by calculating the number of options for each position sequentially. The President must be a senior, so there are 7 choices for President. Once the President is chosen, any of the remaining 11 members can serve as Vice President. After filling the Vice President role, 10 members remain for Secretary. Multiplying these independent choices gives 7×11×10=7707 \times 11 \times 10 = 770.

Step-by-Step Solution

1
Determine the available choices for the President position
7 possibilities
The President position is restricted to seniors only, and there are 7 seniors in the club.
2
Determine the available choices for the Vice President position
11 possibilities
After 1 member is chosen as President, 11 of the original 12 members remain available for the Vice President role.
3
Determine the available choices for the Secretary position
10 possibilities
After 2 members are chosen for President and Vice President, 10 members remain available for Secretary.
4
Calculate the total number of distinct outcomes using the Fundamental Counting Principle
770 total arrangements
Multiplying the choices for each position gives 7×11×10=7707 \times 11 \times 10 = 770.

Key Concept

Fundamental Counting Principle with Position Restrictions
Estimated Time:1m 15s
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Topics

Basic Probability and Counting Methods

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