[Math] Number of combinations from 3 sets.

combinatorics

Lets say I am given 3 sets:

$S_1 = \{a_1, a_2, a_3,…,a_{10}\}\quad$
$S_2 = \{b_1, b_2, b_3,…,b_8\}\quad$
$S_3 = \{c_1, c_2, c_3,…,c_5\}$

I am unsure how to find the total number of combinations if I am supposed to choose 2 elements from $S_1$, 2 elements from $S_2$ and 1 element from any of the sets. I am also unable to choose the same element twice in any combination.

An example of a combination could be $\{a_1, a_3, b_3, b_8, c_2\}$ where there is always 5 elements total out of 23 total elements.

My original answer was incorrect.

Any ideas?

Best Answer

  • Choose two of $10$ from set $S_1$ ($10$-choose-$8$).
  • $\times$
  • Choose two of $8$ from set $S_2$ ($8$-choose-$2$).
  • $\times$
  • For the fifth element, we choose $1$ from the $23 - 4 = 19$ remaining elements, knowing $19$-choose-$1$ = $19$ choices.

$$\binom{10}{2}\times \binom{8}{2}\times [(23-4)]= \binom{10}{2}\times \binom{8}{2}\times (19) = \quad?$$

Recall: $$n-\text{choose}-k = \binom{n}{k} = \dfrac{n!}{k!(n-k)!}$$

Note: we need both "combinations": n-choose-k, or and the "rule of the product", to solve this problem. Both these tools need to become very familiar to you.

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