BTGModeratorVI wrote: ↑Sun Jul 19, 2020 1:33 pm
5 people including A and B line up in a row. How many possible cases are there such that at least one person stands between A and B?
A. 24
B. 36
C. 48
D. 60
E. 72
Answer:
E
Source: Math Revolution
Here's a different approach:
Let's let the 5 people be A, B, C, D and E
Take the task of arranging the 5 people and break it into
stages.
Stage 1: Arrange C, D and E in a row
We can arrange n items in n! ways.
So, we can arrange 3 people in 3! (aka 6) ways.
We can complete stage 1 in
6 ways
IMPORTANT: For each of the 6 arrangements of C, D and E, we can place a space on either side of each person.
For example, for the arrangement DEC, we can add spaces as follows:
_D_E_C_
We can now place A in one of these spaces, and place B in one of these spaces.
This will ensure that A and B do
NOT sit together
Stage 2: Place person A in one of the spaces.
There are 4 spaces to choose from. So, we can complete stage 2 in
4 ways
Stage 3: Place person B in one of the spaces.
There are 3 spaces remaining.So, we can complete stage 3 in
3 ways
By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus arrange all 5 people) in
(6)(4)(3) ways (= 72 ways)
Answer: E
