Question

Ten weight lifters are competing in a team weight-lifting contest. Of the lifters, 3 are from the United States, 4 are from Russia, 2 are from China, and 1 is 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 competitor in the top three and 2 in the bottom three?

Answer 1

(a) 65536 - the basic principle of counting

(b) 2520 - use the multinomials.

Explanation:

Divide 8 people among 4 schools

The teachers are different among themselves, and so are the schools.

(a)

Every teacher can choose a school. By the basic principle of counting: 4.4.4.4.4.4.4.4 = 4^8 = 65536

different divisions of the teachers

Note: this does not ensure that every school gets a teacher

(b) - Every school gets precisely 2 teachers:

to divide 8 different objects into four groups of fixed sizes 2, 2, 2 and 2, use the multinomial coefficient

( 8 2, 2, 2, 2) = (8!)/(2!2!2!2!) ==> 2520

different divisions of the teachers