1. (a) How many ternary strings (strings of [0, 1,2)) of length 5 are there? (b)

Question

1. (a) How many ternary strings (strings of [0, 1,2)) of length 5 are there? (b) How many ternary strings of length 5 are there that does not contain the substring 012? (Thus 20121 is not allowed, but 02102 is allowed, hecause the 0, 1 and 2 are not consecutive.) 2. How many triangles are formed by n lines with no two lines parallel and no three lines intersecting in the same point? 3. How many ways are there to distribute (a) 12 different books among 25 people? (b) 12 identical books among 25 people? (e) 12 different books atnong 25 people if no person gets more than one book? (d) 12 identical books among 25 people if no person gets more than oie book? 10people at a party. How many ways are there to pnir them offinto five 2-player

Explanation / Answer

Ans 4) 10 people to line up such that the first person and the second person make up the first pair, the third and fourth make up the second pair, etcetera. This gives 10! pairings.

However, if the first person would swap places with the second person, the pairing would be no different, though we counted it as another pairing! In fact, each pair can have its members swap places while preserving the pairings, so we counted each pairing 2^5 times. So, that's what we have to divide by.

We're not finished yet, though: what if the first two people swap places with the third and fourth? It would once again preserve the pairings, though we counted it as a different pairing. In fact, the 5 pairs can swap places with each other all they want, the same pairings would still be preserved. Hence we counted each pairing 5! times, and that's what we have to divide by as well.

In conclusion, there are

10!/2^5*5!=945

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