# Combination Question

#### Swazination

##### New member
How many combinations can I make if I have 4 letters A,G,C,T and I want to make groups of 5. Yes, you can repeat the letters.

#### Denis

##### Senior Member
From AAAAA to TGCAA
or (if replaced by digits 1 to 4):
from 1111 to 43211

Yes?

#### ksdhart2

##### Full Member
Combinations/Probability problems of this sort almost always come down to pattern recognition. In fact, learning how to find a pattern and extrapolate what comes next will help you greatly in nearly every aspect of mathematics. So let's try using those skills and see what we can come up with for this problem. Let's start with a much simpler version and build up as we go along. Suppose we had two letters (A and G) and one slot. How many combinations would there be? Well, obviously the answer is two. That's not very interesting nor does it seem helpful just yet, but we'll keep in mind all the same.

Now suppose we had two letters and two slots. How many combinations would there be then? Thinking about it, we can force the first slot to be A, then there's two possibilities for the second slot. Similarly, if we force the first slot to be G, we have two additional combinations. So we have two possibilities for the first slot, and for each of those two possibilities, we have two possibilities for the second slot. That gives us a total of 2 * 2 = 4 = 22 combinations.

Now suppose we had two letters and three slots. How many combinations would there be then? I see that we can arbitrarily fix the first two slots and then let the third one vary. Since we already know that there's four possible ways to arrange the first two slots, and for each of those four possibilities, there's two choices for the third slot, that gives us a total of 4 * 2 = 8 = 23 combinations. Are you seeing a pattern? How many combinations would there be with four slots? Five slots? n slots?

Now let's suppose we had three letters (A, G, and C) and two slots. By the exact same logic as before, we have three choices for the first slot, and for each of those three choices, we have three choices for the second slot. That gives us a total of 3 * 3 = 9 = 32 combinations. How many combinations would there be with three letters and three slots? Four slots? n slots? By this point, you should definitely be starting to see the bigger pattern and know the full answer.

#### pka

##### Elite Member
How many combinations can I make if I have 4 letters A,G,C,T and I want to make groups of 5. Yes, you can repeat the letters.
In the post you use the term combinations which do not allow for repetitions . What it seems you want is a multi-set.
NOTE that order does not matter.
If we want the number of ways to put $$\displaystyle \bf{N}$$ identical items (in this case five choices) into $$\displaystyle \bf{k}$$ distinct cells (in this case four letters) is
$$\displaystyle \dbinom{N+k-1}{N}$$
(in this case $$\displaystyle \dbinom{5+4-1}{5}=\dfrac{8!}{5!\cdot 3!})$$

If you mean that order does matter then it is simply $$\displaystyle \bf 4^5$$

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