Understanding solution to the probability problem from Sheldon Ross' book

Question

I came across following problem from Sheldon Ross' book:

A closet contains 10 pairs of shoes. If 8 shoes are randomly selected, what is the probability that there will be

1. no complete pair;
2. exactly one complete pair

I solved them as follows:

Problem 1

• 1st shoe can be anyone out of 20: $\frac{20}{20}$
• 2nd shoe can be anyone out of remaining 19 except the one forming pair with previously selected shoe: $\frac{18}{19}$
• 3rd shoe can be anyone out of remaining 18 except those forming pair with previously selected shoes: $\frac{16}{18}$

and so on till we choose 8 shoes. We need to take multiplication of all, leading to $\frac{20\times18\times16\times14\times12\times10\times8\times6}{20\times19\times18\times17\times16\times15\times14\times13}=0.091$

Problem 2

Following same logic of problem 1,

• 1st shoe can be any one out of 20: $\frac{20}{20}$
• 2nd show should be the one forming pair with 1st one
• 3rd shoe can be any one out of remaining 18: $\frac{18}{18}$
• 4th show can be any one out of remaining 17 except the one forming pair with previously selected one: $\frac{16}{17}$

and so on till we choose 8 shoes. We need to take multiplication of all, leading to $\frac{20\times1\times18\times16\times14\times12\times10\times8}{20\times19\times18\times17\times16\times15\times14\times13}=0.015$

It turns out that the solution to first problem is correct, but the solution to second problem is incorrect. Its given as follows: $\frac{\binom{10}{1}\binom{9}{6}\color{red}{\frac{8!}{2!}}2^6}{20\times19\times18\times17\times16\times15\times14\times13}$

I understand we can select one pair out of 10 in $\binom{10}{1}$ ways. We select both shoes from this pair. Then we can select six pairs out of remaining nine in $\binom{9}{6}$ ways. We can select any one out of two shoes of each of six pairs in $2^6$ ways. But I dont understand from where $\frac{8!}{2!}$ came.

Edit

The solution given at the back of the book is $0.4268$. But books solution manual solves it as $\frac{\binom{10}{1}\binom{9}{6}\color{red}{\frac{8!}{2!}}2^6}{20\times19\times18\times17\times16\times15\times14\times13}$ which I just checked to be equal to $0.2133$. This pdf gives the solution as $\frac{\binom{10}{1}\binom{9}{6}\times 2^6}{\binom{20}{8}}$ which matches with $0.4268$. So now I am guessing what is correct answer and how can I get the answer for problem 2 by following same approach as I followed for problem 1.

Answer 1

Your text is in error.     There should be no $8!/2!$ factor.

In general the probability for selecting $x$ pairs $(x\in \{0,1,2,3,4\})$ is$$\dfrac{\dbinom{10}{x}\dbinom{10-x}{8-2x}2^{8-2x}}{\dbinom{20}8}\quad\text{or}\quad\dfrac{\dbinom{10}{8-x}\dbinom{8-x}{x}2^{8-2x}}{\dbinom{20}8} $$

The series sums to 1 as required.

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Thus the probability for selecting no pairs is $\binom{10}{8}2^8/\binom{20}{8}$, which is equal to your answer.

And the correct probability for selecting exactly one pair is $\binom {10}{1}\binom{10-1}{8-2}2^{8-2}/\binom{20}{8}$.   Alternatively that is $\binom{10}{8-1}\binom{8-1}12^{8-2}/\binom{20}8$.

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By your method you can select six from the 10 pairs, one shoe from each of those, and both shoes of one from the remaining pair.

$$\dfrac{20\cdot18\cdot 16\cdot14\cdot12\cdot 10\cdot 4/6!}{20\cdot 19\cdot 18\cdot 17\cdot 16\cdot 15\cdot 14\cdot 13}$$

Answer 2

This is a problem similar to deck cards and hands problems.

The twenty shoes may be seen as a deck of two color (Left and Right) and 10-suits.

A "hand" contains 8 cards. It may contain 0 pairs, 1 pair... to 4 pairs, and we are asked to evaluate the chances to get some of these types.

the total number of hands is $ \binom {20}{8}$

i) for no pairs, we have $\binom {10}{8} $ for the choice of the 8 distinct values and $2^8$ for the eight color choices (left or right). The probability of such a "hand" is then 0.0914

ii) for one pair, we have to choose it among ten possibles. Then we have to chose another 6 singles having distinct values from 9 values that remained. Then we have 2^6 choices for the color, for a total $\binom {10} {1} \binom {9}{6} 2^6$. The probability to get a hand like this is 0.4267