Discussion 10
Week of March 31, 2003 Examples
Example 1: One hundred tickets numbered 1, 2, 3, ..., 100, are sold to 100 different
people for a drawing. Four different prizes are awarded, including a grand prize. How
many ways are there to award the prizes if
a) If there are no restrictions they can be awarded in 100·99·98·97 ways.
[or P(100, 4) ways]
b) If the person holding ticket 47 wins the grand prize, there are 1·99·98·97 ways.
[or P(99, 3) ways]
c) If the person holding ticket 47 wins a prize, there are 4·99·98·97 ways
d) If the person holding ticket 47 doesn't win a prize, there are 99·98·97·96 ways
e) If people holding tickets 19 and 47 both win, there are 4·3·98·97 ways since 19 may
win
any of the four prizes, 47 wins one of the remaining three prizes, and two of the other 98
ticket holders win the other two prizes.
f) If people holding tickets 19, 47, and 73 all win, there are 4·3·2·99 ways
g) If people holding tickets 19, 47, 73 and 97 all win, there are 4·3·2·1 ways
h) If none of the people holding tickets 19, 47, 73 and 97 wins, there are 96·95·94·93
ways
i) If the grand prize winner is a person holding ticket 19, 47, 73, or 97, there are
4·99·98·97 ways
j) If people holding 19 and 47 win prizes and those holding 73 and 97 do not, there are
4·3·96·95
ways.
Example 2: A committee is chosen from a group containing 4 men and 6 women.
a) A committee of size three can be found in C(10, 3) = (10·9·8) /3! ways.
b) A committee of size three containing 1 man and 2 women can be found in
C(4,1)·C(6,2) or 60 ways.
Stress that the order in which the members are chosen is irrelevant.
c) A committee with two men and three women can be found in C(4, 2)·C(6, 3) = 120
ways.
d) A committee of size 4 with at least as many women as men can be found in
C(4,0)·C(6,4) + C(4, 1)·C(6,3)
+ C(4,2)·C(6,2)
ways.
all women
+ 1 man and 3 women + 2 men and 2 women.
Example 3: Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find the number of subsets of S that
a. contain the number 5--29
b. contain only even numbers -- 25
c. contain exactly three elements, one of which is 3—C(9, 2)
d. contain exactly five elements, but neither 3 nor 4—C(8, 5)
Example 4: How many positive integers larger than 5,000,000 can be obtained using
3,4,4,5,5,6, and 7 as the digits?
Example 5: How many four digit numbers contain
a. at least one even digit ? 9000 - 54
b. exactly two even digits if no digits are repeated
even in first position: 4 choices for the even digit in first position
3 places for second even digit
4 choices for second even digit
5·4 ways to put in the odd digits
Total: 3·43·5
odd in first position: 5 choices for the odd digit in first position
3 places for second odd digit
4 choices for second odd digit
5·4 ways to put in the even digits
Total: 3·42·52
The answer is the sum of these two values.
Example 6: There are 10 rooms in a row and a party of 10 people are put into the rooms,
one per room. In how many ways can this be done if A and B must occupy
adjacent rooms.
Answer: Treat this as an AB pair and 8 other people giving 9! ways. However, we
could also have BA to satisfy the condition so the final answer is 2•9!