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# coin + probability

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i have 8 pennies, 5 quarters, and 3 nickels in my wallet. if i take them out of my wallet one at a time without replacement, what is the probability that the eighth coin is a nickel, and the twelfth is a quarter?

May 20, 2021

#1
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There are $$\frac{16!}{3!5!8!}$$different arrangements of the 16 coins. If you fix the 8th and 12th spots as being occupied by by a nickel and a quarter, then you have only 14 coins left to arrange and there are $$\frac{14!}{2!4!8!}$$possibilities for the remaining coints. So the probability we are looking for is$$\frac{\frac{14!}{2!4!8!}}{\frac{16!}{3!5!8!}}$$, which I think simplifies to $$\frac{1}{16}$$.To see what is happening, let's look at a simpler problem:

Suppose you only have 1 nickel (N), two quarters(Q), and 3 pennies(P). These coins can be arranged in 60 different ways ($$\frac{6!}{3!2!1!}$$).

The following diagram show the sixty different arrangements; of those 12 of them have a quarter in the 3rd and a penny in the 5th spots. so the probability of drawing a quarter on the third and a penny on the 5th draw is $$\frac{12}{60}=\frac{1}{5}$$.

May 20, 2021
#2
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Nice work guest  :)

Melody  May 20, 2021