Answer to Question #105082 in Statistics and Probability for Shuvo

Question

Exercise 1:

A young boy asks his mother to get 5 picture card from his collections of 10 flower picture cards and 5 sports picture cards. How many are there that his mother can get 3 flower and 2 sports picture cards?

Exercise 2:

A present and a treasurer are to be chosen from a student club consisting of 50 people. How many different choices of officers are possible if (1) there are no restrictions; (2) A will serve only if he is a president; (3) B and C will serve together or not at all; (4) D and E will not serve together?

Answer 1

Exercise 1: 

$$\binom{10}{3}\binom{5}{2}={10! \over 3!(10-3)!}\cdot{5! \over 2!(5-2)!}=$$

$$={10\cdot9\cdot8 \over 1\cdot2\cdot3}\cdot{5\cdot4 \over 1\cdot2}=1200$$

There are 1200 ways his mother can get 3 flower and 2 sports picture cards 

Exercise 2: 

(1) There are no restrictions

$$P(50,2)={50! \over (50-2)!}=50\cdot49=2450$$

(2) A will serve only if he is a president

A will serve

$$\binom{49}{1}={49! \over 1!(49-1)!}=49$$

A will not serve

$$P(49,2)={49! \over (49-2)!}=49\cdot48=2352$$

How many different choices of officers are possible?

$$49+2352=2401$$

Or how many different choices of officers are possible?

$$2450-49=2401$$

(3) B and C will serve together or not at all

B and C will serve together: $2!=2$

If B will not serve and C will not serve

$$P(48,2)={48! \over (48-2)!}=48\cdot47=2256$$

How many different choices of officers are possible?

$$2+2256=2258$$

Or how many different choices of officers are possible?

$$2450-2\cdot48-2\cdot48=2258$$

(4) D and E will not serve together

If D will be a president: $48$.

If D will be a treasurer: $48.$

If E will be a president: $48.$

If D will be a treasurer: $48.$

How many different choices of officers are possible?

$$2\cdot48+2\cdot48=192$$