Question

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

1) There are no restrictions:

For the first position we have 50 choices and 49 for the second, thus:

50 * 49 = 2450

or $P(50,2)= \frac{50!}{(50−2)!} =50⋅49=2450$

2) A will serve only if he is a president

There two possibilities:

A will serve: $C(49,1)= \frac{49!}{1!(49−1)!} =49$

A will not serve: $P(49,2)= \frac{49!}{(49−2)!} =49⋅48=2352$

thus 49 + 2352 = 2401

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

B and C will serve together: $2!=2$

B will not serve and C will not serve: $P(48,2)= \frac{48!}{(48−2)!} =48⋅47=2256$

thus 2 + 2256 = 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

thus 48 + 48 + 48 + 48 = 192

Answer:

1) 2450

2) 2401

3) 2258

4) 192