Answer to Question #282373 in Discrete Mathematics for Joe

Question

Out of a group of ten residents in a certain county, 3 are Republicans, 5 are

Democrats, and 2 are Independents. How many unique partitions of this group

of residents are there by political party?

38) You have eight distinct pieces of food. You want to choose three for breakfast,

two for lunch, and three for dinner. How many ways to do that?

39) You teach a class with 90 students. In a rather severe effort to combat grade

inflation, your department chair insists that you assign the students exactly 10

A’s, 20 B’s, 30 C’s, 20 D’s, and 10 F’s. How many ways to do this?

40) How many 13-card bridge hands have 4 of one suit, 3 of one suit, 5 of one suit, 1

of one suit?

Answer 1

Solution:

(37):

We can simply plug in the following values into the formula for the multinomial coefficient:

n (total residents): 10

n₁ (total Republicans): 3

n₂ (total Democrats): 5

n₃ (total Independents): 2

Multinomial Coefficient = 10! / (3! * 5! * 2!) = 2,520

(38):

No. of ways $=\dfrac{8!}{3!2!3!}=560$

(39):

No. of ways $=\dfrac{90!}{10!20!30!20!10!}$

(40):

No. of ways $=^{13}C_4\ ^{13}C_3 \ ^{13}C_5\ ^{13}C_1=3421322190$