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|||Permutations and combinations: Problem type 1Suppose we want to choose 4 colors, without replacement, from 17 distinct colors.(a) How many ways can this be done, if the order of the choices matters?(b) How many ways can this be done, if the order of the choices does not matter?**

Respuesta :

Answer:

a) Number of ways to choose 4 colors from 17 if the order matters =57120

b) Number of ways to choose 4 colors from 17 if the order does not matter = 2380

Explanation:

Total number of colors, n = 17

Number of colors to choose, k = 4

a) Number of ways to choose 4 colors from 17 if the order matters = 17P4

[tex]\begin{gathered} nPk=\frac{n!}{(n-k)!k!} \\ \\ 17P4=\frac{17!}{(17-4)} \\ \\ 17P4=\frac{17!}{13!4!} \\ \\ 17P4=\frac{17\times16\times15\times14\times13!}{13!} \\ \\ 17P4=17\times16\times15\times14 \\ \\ 17P4=57120 \end{gathered}[/tex]

b) Number of ways to choose 4 colors from 17 if the order does not matter = 17C4

[tex]\begin{gathered} nCk=\frac{n!}{(n-k)!k!} \\ \\ 17C4=\frac{17!}{(17-4)!4!} \\ \\ 17C4=\frac{17!}{13!4!} \\ \\ 17C4=\frac{17\times16\times15\times14\times13!}{13!\times4\times3\times2\times1} \\ \\ 17C4=2380 \end{gathered}[/tex]

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