# Permutation and Combination

#### guest17

##### New member

how many different combinations are there of 12 things, four of which are alike, if five are taken at a time?
I approached this question by calculating the possibilities with no restrictions, then dividing the 4 repeated items. I got (12C5)/4!...

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#### pka

##### Elite Member
how many different combinations are there of 12 things, four of which are alike, if five are taken at a time?
$$\displaystyle \[\sum\limits_{k = 0}^4 \dbinom{8}{5-k}$$ HOW/WHY?

#### soroban

##### Elite Member
Hello, guest17!

I see no formula for this problem . . .

How many different combinations are there of 12 things,
four of which are alike, if five are taken at a time?

We have: .$$\displaystyle \{A, A,A,A, B,C,D,E,F,G,H,I\}$$
. . four A's and eight Others.

$$\displaystyle \begin{array}{ccccc} \text{0 A, 5 Others:} & _8C_5 &=& 56 \\ \text{1 A, 4 Others:} & _8C_4 &=& 70 \\ \text{2 A, 3 Others:} & _8C_3 &=& 56 \\ \text{3 A, 2 Others:} & _8C_2 &=& 28 \\ \text{4 A, 1 Others:} & _8C_1 &=& \;8 \\ \hline \text{Total:} &&& 218\; \end{array}$$

Thank you!