Question

There are 8 people at a picnic and their ages are as follows 15, 5, 2, 20, 7, 30, 40, 23. A group of 3 people is picked at random.

a) How many groups are possible?

b) How many groups are there with a combined age greater than 27?

Answer 1

The order with which the people are picked doesn't seem to matter, so you use the combination formula:
\[\large_{8}C_{3}=\frac{8!}{3!(8-3)!}=56\]

Answer 2

I'm sure there's a quick way to figure out part (b), but so far I can think of it.

Count up all the combinations of three of the numbers that sum to something greater than 27.
\[2,~5,~7,~15,~20,~23,~30,~40\]
2, 5, 23
2, 5, 30
2, 5, 40
2, 7, 20
2, 7, 23
2, 7, 30
2, 7, 40
and so on.

Answer 3

thanks so much

Answer 4

I suppose a quick way is to count all the groups of people that already contain someone older than 27.

We'd then have
\[2\cdot\large_{7}C_{2}=2\cdot21=42\]
because any group containing the 30 or 40 year old will automatically qualify as "greater than 27."

Add up the remaining combinations that qualify.