Question

A group of 75 friends meets for lunch. They greet each other by exchanging fist bumps. How many fist bumps are exchanged if each friend must bump with each of the 74 ​others?

Answer 1

There is a group of 75 friends and each friend must bump with each of the 74 others.
So it means that we have to multiply 75 by 74.
75 * 74 = 5,550
Answer:
5,550 first bumps are exchanged.

Answer 2

we know that

The first person needs to fist bump all the other people

$(75 - 1 (themselves) = 74)$

The second person needs to fist bump everyone except the first person

$(75 - 1 (themselves) - 1 (first\ person\ already\ bumped\ them) = 73$

and so on until the fist bump fest is over

So

$74+73+72+71+70+....+3+2+1$

To sum up the terms of this arithmetic sequence, use this formula:

$S=\frac{n}{2} (2a+(n-1)d)$

where

a is the first term

d is the common difference between terms

n is the number of terms to add up

in this problem

$a=74\\ n=74\\ d=-1$

Substitute the values in the formula above

$S=\frac{74}{2}*(2*74+(74-1)*(-1)) \\ \\ S=\frac{74}{2}*(148-73)\\ \\ S=\frac{74}{2} *75\\ \\ S=2,775$

therefore

the answer is

$2,775$