Question

Suppose we want to choose 4 colors, without replacement, from 18 distinct colors.

Answer 1

Answers:

• (a) 73440
• (b)  3060

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Explanation for part (a)

Consider four slots labeled A,B,C,D.

We have

• 18 choices for slot A
• 17 choices for slot B
• 16 choices for slot C
• 15 choices for slot D

We started at 18 and counted down until we filled the four slots. The countdown is because we cannot repeat a color. Multiply out those values to get the answer: 18*17*16*15 = 73440

You can use the nPr permutation formula as an alternative method

$_nP_r = \frac{n!}{(n-r)!}$

in this case n = 18 and r = 4. The exclamation marks indicate factorial.

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Explanation for part (b)

Let's say we had color labels A,B,C,... all the way up to R which is the 18th letter in the English alphabet. From those 18 letters, we can only pick four. Let's say we pick D,E,F,G.

Focusing solely on the set {D,E,F,G}, we only have one set and the order doesn't matter. So {D,E,F,G} is the same as {D,E,G,F}.

There are 4*3*2*1 = 24 ways to arrange those four items meaning that we'll need to divide the result of part (a) by 24 to get the result of part (b)

73440/24 = 3060

You could use the nCr combination formula to get the same result

$_nC_r = \frac{n!}{r!(n-r)!}$

where n = 18 and r = 4 are the same from last time.

The connection between nPr and nCr is this

$_nC_r = \frac{_nP_r}{r!}$

which can be rearranged into

$_nP_r = r!*\left( _nC_r \right)$

these formulas are handy to help us go back and forth between nCr or nPr.