Question

A president and a treasurer are to be chosen from a student club consisting of 100
people. How many different choices of officers are possible if
a. Julian and Anton will serve together or not at all?
b. There are no restrictions?
c. Julian will serve only if she is president?
d. Sara and Silvia will not serve together?
e. Note that there are no repeated names in this student club.

Answer 1

You need to chose two positions from 100 people.

a. Julian and Anton will serve together or not at all:

If Julian and Anton serve together $2!=2$

If Julian and Anton do not serve we have 98 choices for the 1st position, we have 97 choices for the 2nd.

$2+98\cdot97=9507$

or

$2!+P(98, 2)=2!+\frac{98!}{(98-2)!}=2+\frac{98!}{96!}=2+97\cdot98=9507$

b. There are no restrictions:

For the 1st position we have 100 choices, for the 2nd we have 99 choices.

$100\cdot99=9900$

or

$P(100,2)=\frac{100!}{(100-2)!}=\frac{100!}{98!}=100\cdot99=9900$

c. Julian will serve only if she is president:

If Julian serves her position is already chosen, thus there are 99 choices for 2nd position.

If Julian does not serve for the 1st position we have 99 choices, for the 2nd we have 98 choices.

$99+99\cdot98=9801$

or

$C(99,1)+P(99, 2)=\frac{99!}{1!(99-1)!}+\frac{99!}{(99-2)!}=\frac{99!}{98!}+\frac{99!}{97!}=99+98\cdot99=9801$

d. Sara and Silvia will not serve together:

If Sara serves there are 2 choices for her position and 98 choices for 2nd position.

If Silvia serves there are 2 choices for her position and 98 choices for 2nd position.

$2\cdot98+2\cdot98=392$

or

$2!\cdot C(98, 1)+2!\cdot C(98, 1)=2\cdot2!\cdot \frac{98!}{1!(98-1)!}=2\cdot2\cdot \frac{98!}{97!}=4\cdot 98=392$

Answer:

a. 9507

b. 9900

c. 9801

d. 392