Combinatorics practice problems - 12,14,15 pending

Ex 11: Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta, and Gamma. The three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can professors Alpha, Beta, and Gamma choose their chairs?

Answer: 60

Solution:
First put chairs of six students. Now there are 5 slots to choose from.
S1S2S3S4S5S
5C3 * 3! = 60

Ex 13. Three integers are selected from the set S = 1, 2, 3, ... , 19, 20. If the number of selections where the sum of the three integers is divisible by 3 is x, then find last two digits of x.
Answer: 84

Solution:
1. Choose all 3 numbers with remainder 0 by 3.
2. Choose all 3 numbers with remainder 1 by 3.
3. Choose all 3 numbers with remainder 2 by 3.
4. Choose 1 from each group.

6C3  + 7C3 + 7C3 + 6C1.7C1.7C1 = 384














Ex 16:
At a workshop, there are 27 participants. Each of them shakes hand once with one another. If total number of handshakes are X. Find the last two digits of X.

Answer: 27C2, then compute.








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