Answer:
there were a total of 21 handshakes that took place in the room.
Step-by-step explanation:
To find the total number of handshakes that took place in the room, we can use the combination formula.
Given that there are 7 members on the board of management, and each member shakes hands with every other member, we need to find the total number of combinations of 2 members out of 7. This will give us the total number of handshakes.
The combination formula is given by:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
Where:
- \( n \) is the total number of items (in this case, the total number of members),
- \( k \) is the number of items to choose (in this case, 2 members for a handshake),
- \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).
Substituting the values into the formula:
\[ C(7, 2) = \frac{7!}{2!(7-2)!} \]
\[ C(7, 2) = \frac{7!}{2!5!} \]
\[ C(7, 2) = \frac{7 \times 6}{2 \times 1} \]
\[ C(7, 2) = \frac{42}{2} \]
\[ C(7, 2) = 21 \]
So, there were a total of 21 handshakes that took place in the room.
