Midterm #2 answers (questions have been paraphrased) 1. What's the probability of getting a straight dealt in order? A. The first card can be a 2, 3, . .., 10. That's 9 possibilities. Then each of the five cards can be any suit (but we know what the ranks must be). That's another factor of 4^5. Then we divide by the number of ways of being dealt 5 cards, which is 52 * 51 * 50 * 49 * 48. 2. How many strings with k 1s, no 000, and beginning and ending with 0? A. Put in the k 1s first. Now put in 0s. At the beginning you could have either "0" or "00" before the first 1, likewise at the end, so that's 2*2 choices. Then in between each pairs of 1s (and there are k-1 such gaps) you can have "00", "0", or "". That's 3^(k-1) choices. So the total is 4 * 3^(k-1). 3a. What's the biggest a_n could be in a slow-growing sequence? A. Well, a_n is bounded by a_(n-1) + n, which is bounded by a_(n-2) + (n-1) + n, . .., which is bounded by 1 + 2 + . .. + n. That's an okay answer, though summing it and getting [n+1 choose 2] is much nicer.
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