Question

Ten weight lifters are competing in a team weight-lifting contest. The lifters $3$are from the United States,$4$are from Russia, $2$are from China, and $1$are from Canada. If the scoring takes account of the countries that the lifters represent, but not their individual identities, how many different outcomes are possible from the point of view of scores? How many different outcomes correspond to results in which the United States has $1$competitors in the top three and $2$in the bottom three?

Short answer

$\left(a\right)12600$possible rankings - by the use of the multinomial coefficients

$\left(b\right)945$possible rankings - the basic principle of counting and multinomial coefficients.

Step-by-step solution

Given Information.

Ten weight lifters are competing in a team weight-lifting contest. The lifters$3$are from the United States,$4$are from Russia, $2$are from China, and$1$ are from Canada. If the scoring takes account of the countries that the lifters represent, but not their individual identities.

Part (a) Explanation.

permutation where only these objects are permuted.

Generalized, this number is called the multinomial coefficient.

Here, there are $10$elements, which$3,4,$$2$are considered the same themselves. So there are$\left(\begin{array}{c}10\\ 3,4,2,1\end{array}\right)-\frac{10!}{3!4!2!1!}-12600$possible rankings.

Part (b) Explanation.

The US has $1$competitors at the top $3$and $2$at the bottom $3$of counting:$3·3·\left(\begin{array}{c}7\\ 4,2,1\end{array}\right)-9·\frac{7!}{4!2!1!}-945$possible rankings.