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.

9987

step-by-step explanation: