A president and a treasurer are to be chosen from a student club consisting of 50
people. How many different choices of officers are possible if
(a) there are no restrictions;
(b) A will serve only if he is president;
(c) B and C will serve together or not at all;
(d) D and E will not serve together?
Answers
Answer:
Answer:
1) 2450
2) 2401
3) 2258
4) 192
Step-by-step explanation:
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=2450P(50,2)=
(50−2)!
50!
=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)!} =49C(49,1)=
1!(49−1)!
49!
=49
A will not serve: P(49,2)= \frac{49!}{(49−2)!} =49⋅48=2352P(49,2)=
(49−2)!
49!
=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!=22!=2
B will not serve and C will not serve: P(48,2)= \frac{48!}{(48−2)!} =48⋅47=2256P(48,2)=
(48−2)!
48!
=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