Ways of making 10 letter combinations out of 3 given letters

combinationscombinatorics

Find the number of 10 letter combinations from ${a, b, c}$ that contain:

  1. At least one of each letter
  2. At least two of each letter
  3. At least one A, two Bs, three Cs
  4. Any number of each letter

(Also repetition is allowed)

For number 1 I think that the answer is $\binom{9}{2}$, since there's n-1 places that I can place a bar from stars&bars.
For number 2 though I'm not sure how I can apply this in order to place the bars so that I get at least 2 of each element $a,b,c$.

Edit 2: I did some thinking about question 2 and I think I might have an answer which is correct. Since I need at least 2 of each letter I decided to group 1 of $AA, BB, CC$ so that $AA$ was one object meaning that now in total I had 7 objects. Which then means that I would have $\binom{6}{2}$ places to divide this group up. Please let me know if this approach is correct. Or if I've made a mistake. Thank you.

Best Answer

Let frequency of $a,b,c$ be $x,y,z$ respectively. You're looking for integral solutions of following cases :

  1. $x \ge 1, y \ge 1, z \ge 1$ and $x+y+z=10$
  2. $x \ge 2, y \ge 2, z \ge 2$ and $x+y+z=10$

This is same as $u \ge 1, v \ge 1, w \ge 1$ and $u+v+w=7$

  1. $x \ge 1, y \ge 2, z \ge 3$ and $x+y+z=10$

This is same as $x \ge 1, p \ge 1, q \ge 1$ and $x+p+q=7$

  1. $x \ge 0, y \ge 0, z \ge 0$ and $x+y+z=10$

Hence answers would be $\binom{9}{2}, \binom{6}{2},\binom{6}{2}, \binom{12}{2}$ respectively.

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