Alexandra Richardson

2022-07-16

How many ternary strings of length 10 that contain exactly 2 1s and 3 2s. Since it must contain exactly 2 1s and 3 2s the other 5 spaces must be 0s.But how many ways can we arrange this?

Wayne Everett

Expert

Step 1
$\left(\genfrac{}{}{0}{}{2}{10}\right)×\left(\genfrac{}{}{0}{}{3}{8}\right)=\frac{10!}{2!\left(10-8\right)!}\frac{8!}{3!\left(8-3\right)!}=\frac{10!}{2!3!5!}$
(using classical formula $\left(\genfrac{}{}{0}{}{n}{p}\right)=\frac{n!}{p!\left(n-p\right)!}$.)
Step 2
You first choose the 2 places (among 10) where you place the 1s, then, for each of these choices, you can select the 3 places where you will place a 2 (in each case among the remaining 8 places). No more choices for the other places where the zeros will be placed.

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