There are 10 students in a class. A student is allowed to shake hand only once with a student who is taller than him or equal in height to him. He can’t shake hand with anyone who is shorter than him. Average height of the class is 5 feet. What is the difference between the maximum and minimum number of handshakes that can take place in the class?

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Maximum number of handshakes occurs when all students are of the same height. In this case, there are no restrictions, so each student can shake hands with every other student. The number of handshakes is given by the combination of 10 students taken 2 at a time:

10C2 = 45

.

Minimum number of handshakes occurs when all students are of different heights. Since a student can only shake hands with someone taller or of equal height, and no one has the same height, no handshakes are possible. Thus, the minimum number is

0

.…

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What this question tests

What it tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Common trap

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.