8. There are 7 Republican, 5 Democrat, and 3 Independent candidates for city may
ID: 3146492 • Letter: 8
Question
8. There are 7 Republican, 5 Democrat, and 3 Independent candidates for city mayor. A group of 6 candidates will be chosen for a televised debate. a. In how many ways can a group of 6 candidates be chosen? b. How many groups have 3 Republicans and 3 Democrats? c. How many groups have no independent candidates? d. How groups have at least one independent candidate? e. Using your work above, what is the probability that at least one independent candidate will be on the televised debate? 9. Suppose you play a lottery where you have to pick 4 numbers from the range 1 - 40. What is the probability you match exactly 3 of the 4 numbers on your lottery ticket?Explanation / Answer
8. There are 7 Republican, 5 Democrat and 3 Independent candidates.
a. Since there are 15 candidates and we need to choose 6, number of groups
= 15C6 = 5005.
b. The 3 Democrats can be chosen in 7C3 ways and the 3 Democrats in 5C3 ways.
=> Total number of groups = 7C3 * 5C3
= 35 * 10
= 350.
c. Since there are 7+5 = 12 non-independents, the number of groups with no independent candidates
= 12C6 = 924.
d. Number of groups with atleast one independent candidate = 5005 - 924 = 4081.
e. Probability that atleast one independent candidate will be on the group = 4081 / 5005 = 0.8154.
9. Four number can be chosen out of 40 in 40C4 ways.
Of these 3 numbers should match.
The non matching number can be any of the 36 numbers remaining except the right match.
The non matching number can itself be any of the four numbers.
=> Probability that exactly 3 numbers match = 36 * 4 / 40C4
= 144 / 91390
= 0.0016.
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