(2) Each of the 50 US states has determined five candidates it can send to the N
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Question
(2) Each of the 50 US states has determined five candidates it can send to the National Math ematics Bee in Mathis, Texas. (a) In the year 1974, each state must either choose one of the five to send or send nobody. Write an expression encoding the number of ways the states can combine to send 20 candidates (b) In the year 2017, the rules have changed. Now each state may select up to three candidates to send to Texas. Write an expression encoding the number of ways the states can combine to send 20 candidates.Explanation / Answer
a) In 1974, each state had two options: 1) to select any one candidate out of the 5 candidates to send or 2) to send nobody.
So, each state could do this in ( 5C0 +5C1) ways i.e. in 6 ways.
For the total 50 states, the total no. of ways
=6*6*...*6 (50 times)
=650
b) In 2017, each state can send up to three selected candidates. So, they can send none or one or two or three selected candidates.
Each state can do this in (5C0+5C1+5C2+5C3) ways i.e. in 26 ways.
So, totally, 50 states can send the candidates in 26*26*...*26 ways (50 times). Hence the total no. of ways
= 26*26*...*26 (50 times)
=2650
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