How many ways can word BANANA be arranged so that all As come together?

In order to get the word BANANA when rearranging those letters, you'll need to choose a B first, then an A, then an N, etc.

For the first spot, there's 1 possible B to choose. For the second spot, there are 3 possible A's to choose. For the third spot, there are 2 possible N's to choose. For the fourth spot, there are 2 remaining A's to choose. For the fifth spot, there's 1 remaining N to choose. For the sixth spot, there's 1 remaining A to choose. That's a total of:

1×3×2×2×1×1 = 12 ways to get the word BANANA.

You might also want to know how many total ways there are to rearrange those letters.

For that problem, there are 6 possible letters to choose for the first spot, 5 remaining for the second spot, 4 for the third spot, 3 for the fourth spot, 2 for the fifth spot, and 1 for the sixth spot. That's a total of:

6! = 6×5×4×3×2×1 = 720 ways

But we've double-counted a few combinations... BANANA (where the first A is A1) and BANANA (where the first A is A2) are identical. We'll need to divide by the number of ways there are to rearrange just the A's and the number of ways there are to rearrange just the N's, since we want to treat all the A's the same and all the N's the same.

There are 3×2×1 = 6 ways to rearrange the A's and 2×1 = 2 ways to rearrange the N's. So, there are:

720/(6×2) = 720/12 = 60 ways to rearrange all the letters in BANANA.

That gives you a 12/60 = 1/5 probability of actually getting the word BANANA, which is pretty good!