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1. You are playing online Scrabble, and your letters are A, A, E. N. S. T. T. The computer will randomly rearrange your letters for you so you can see possible words to play. [2) (a) Find the total number of unique ways your letters can be rearranged. (3) (b) Find the probability that the three vowels (A, A, E) are all together (in any order). (4] (c) Find the probability that there is a consonant at each end. (3) (d) Find the probability that the letters on each end are different. (21 (e) Suppose you randomly choose 4 letters. Find the probability you choose two consonants and two vowels. 2. Suppose that the lifetime of a certain brand of light bulb follows an exponential distribution with mean lifetime 1500 hours, If the lifetime of a bulb exceeds 4000 hours, it is said to have "great" quality. If the lifetime of a bulb is between 2000 and 4000 hours, it is said to have "good" quality. If the lifetime of a bulb is between 500 and 2000 hours, it is said to have "fair" quality. If the lifetime of a bulb is less than 500 hours, it is said to have "poor" quality. When a box contains 20 light bulbs, what is the probability that: [6] (a) there are 4 bulbs with "great" quality, 6 with "good" quality, 8 with "fair" quality and 2 with "poor" quality? [3] (b) there are 6 bulbs with "good" quality and 6 with "poor" quality? (3) (c) there are 10 bulbs with "fair" quality, given that there are no bulbs with "poor" quality?