Difference between revisions of "2024 DMC Mock 10 Problems/Problem 11"
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First we use complementary counting to count the total number of possibilities. | First we use complementary counting to count the total number of possibilities. | ||
There are <math>4! = 24</math> ways to arrange the officers without restrictions, and <math>2 \cdot 6! = 12</math> ways if the | There are <math>4! = 24</math> ways to arrange the officers without restrictions, and <math>2 \cdot 6! = 12</math> ways if the | ||
| − | treasurer and president sit next to each other, so the officers can sit in a total of <math>24 | + | treasurer and president sit next to each other, so the officers can sit in a total of <math>24 - 12 = 12</math> |
ways. Similarly, there are <math>4</math> ways for both the treasurer and vice president to sit next to the | ways. Similarly, there are <math>4</math> ways for both the treasurer and vice president to sit next to the | ||
president. Therefore, the answer is <math>\frac{12-4}{12}=\boxed{\frac{2}{3}}</math>. | president. Therefore, the answer is <math>\frac{12-4}{12}=\boxed{\frac{2}{3}}</math>. | ||
Latest revision as of 13:10, 22 December 2024
First we use complementary counting to count the total number of possibilities.
There are 
ways to arrange the officers without restrictions, and 
ways if the
treasurer and president sit next to each other, so the officers can sit in a total of 
ways. Similarly, there are 
ways for both the treasurer and vice president to sit next to the
president. Therefore, the answer is 
.
