Answer to Question #348799 in Discrete Mathematics for Joy

Question

From a committee of 10 people,

1. In how many ways can we choose a chairperson, a vice-chairperson, and a secretary, assuming that one person cannot hold more than one position?

2. In how many ways can we choose a subcommittee of 3 people?

3. Find the number of combinations of 13 objects taken 8 at a time.

4. How many 5-card hands will have 3 aces and 2 kings?

5. Serial numbers for a product are to be made using 2 letters followed by 3 numbers. If the letters are to be taken from the first 8 letters of the alphabet with no repeats and the numbers are taken from the 10 digits (0-9) with no repeats, how many serial numbers are possible?

6. A company has 7 senior and 5 junior officers. An ad hoc legislative committee is to be formed. In how many ways can a 4-officer committee be formed so that it is composed of

a. Any 4 officers?

b. 4 senior officers?

c. 3 senior officers and 1 junior officer?

d. 2 senior officers and 2 junior officers?

e. At least 2 senior officers?

Answer 1

1.

$$10(9)(8)=720$$

2.

$$\dbinom{10}{3}=120$$

3.

$$\dbinom{13}{8}=1287$$

4.

$$\dbinom{4}{3}\dbinom{4}{2}=4(6)=24$$

5.

$$\dfrac{8!}{(8-2)!}\cdot \dfrac{10!}{(10-3)!}=8(7)(10)(9)(8)=40320$$

6.

a.

$$\dbinom{7+5}{4}=495$$

b.

$$\dbinom{7}{4}\dbinom{5}{0}=35(1)=35$$

c.

$$\dbinom{7}{3}\dbinom{5}{1}=35(5)=175$$

d.

$$\dbinom{7}{2}\dbinom{5}{2}=21(10)=210$$

e.

$$\dbinom{7}{2}\dbinom{5}{2}+\dbinom{7}{3}\dbinom{5}{1}+\dbinom{7}{4}\dbinom{5}{0}$$

$$=21(10)+35(5)+35(1)=420$$