5students, including alice, bob, and cindy, line up in a row. in how many ways can they line up so that alice is somewhere between bob and cindy ( not necessarily adjacent to them )?

20 possible arrangements

Step-by-step explanation:

A: Alice  

B: Bob

C: Cindy

X: other student

All the possible arrangements are:

When Bob is in the first place

B A C X X

B A X C X

B A X X C

B X A C X

B X A X C

B X X A C

6 possible arrangements

When Bob is in the second place (notice that neither Alice nor Cindy can be before Bob)

X B A C X

X B A X C

X B X A C

3 possible arrangements

When Bob is in the third place

X X B A C

Only 1 possible arrangement.

Total arrangements when Bob is before Cindy: 10

If Cindy is before Bob the same situations are repeated, given another 10 arrangements. So, the total line up in which Alice is somewhere between Bob and Cindy are 20.