A box contains 7 letters, shown as CHARTED. What is the probability of the outcome RAT in that order if 3 letters are drawn one by one (a) with replacement? (b) without replacement? The probability of the outcome RAT in that order if 3 letters are drawn with replacement is \( \frac{1}{343} \). (Type your answer as a fraction.) The probability of the outcome RAT in that order if 3 letters are drawn without replacement is (Type your answer as a fraction.)

Since the letters in CHARTED are all distinct, we have 7 letters. (a) With replacement, each draw is independent. The probability of drawing each letter (R, then A, then T) is 1/7, so the probability for RAT in that order is   (1/7) × (1/7) × (1/7) = 1/343. (b) Without replacement, the probability changes because each draw affects the next. For the first draw, the probability of drawing R is 1/7. After R is removed, there are 6 letters left, and the probability of drawing A is 1/6. After drawing A, 5 letters remain, and the probability of drawing T is 1/5. Therefore, the probability is   (1/7) × (1/6) × (1/5) = 1/210. So, the probability of getting RAT in that order when drawing without replacement is 1/210.