1. Show that if you pick 19 points from the interior of a unit square, then there exist

three points that are all at most V/3 units apart from each other. (Hint: Use the

pigeonhole principle.)

2. How many different 10-letter strings can be made from the letters of MISSISSIPPI,

which is 11 letters? (Hint: Break down into which letter gets left out. The answer

is 34650 but you must explain why.)

3. How many ways can we roll 6 regular dice (think of the dice as being 6 different

colors) where they add up to 12? Note that there is an upper bound on the numbers

on the dice. (hint: The answer is 456 but you must explain why.)

4. You are given a random collection of 7 cards out of the standard deck of 52. Find

the probability of

(a) having at most two spack.

(b) Having “two pair,” that is, 2 each of two different ranks, and 1 card each of

another three ranks. For example, {7c, 75, 4s, 4h, 9e, Jh, 3d}

5. 7.1: 28.

6. 10 dice (all of different colors, say) are rolled.

(a) \Vhat is the probability that there are 4 each of two different numbers, and 1

each of another two numbers? For example. any arrangement of 3333555516.

(b) What if, furthermore, all rolls of the sanie number have to occur consecutively?

For example, the arrangement 1555566663 is allowed, but 6615555366 is not

allowed.

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