Basic quen on probability

Question

Quen: An ant farm contains both red and black ants. A particular passage in the farm is so narrow that only 1 ant can get through at a time. If 4 ants follow each other through the passage, how many different color patterns (having 4 elements) could be produced (assuming that red ants are indistinguishable from one another, as are black ants)?

Answer 1

Since one red ant is indistinguishable from the other red ant then their order doesn't matter, as well as the black ants from the others. Therefore, we can use the combination formula:

_nC_r = n! / (r! (n-r)!)

The combination of n items taken r at a time.

The following patterns can be:

No black (4 red) = RRRR → ₄C₀ = 4! / (0! (4-0)!) = 1

1 black (3 red) = BRRR, RBRR, RRBR, RRRB → ₄C₁ = 4! / (1! (4-1)!) = 4

2 black (2 red) = BBRR, BRBR, BRRB, RBRB, RRBB, RBBR → ₄C₂ = 4! / (2! (4-2)!) = 6

3 black (1 red) = BBBR, BBRB, BRBB, RBBB → ₄C₃ = 4! / (3! (4-3)!) = 4

4 black (no red) = BBBB → ₄C₄ = 4! / (4! (4-4)!) = 1

₄C₀ + ₄C₁ + ₄C₂ + ₄C₃ + ₄C₄ = 1 + 4 + 6 + 4 +1 = 16

There are 16 color patterns.

Another way to solve this is using Pascal Triangle of Numbers.