Question

You have the letters ACTIVE placed in a hat. You pick the letters out one at a time to create a sequence of letters, WhatIs the probability that you spell a word in which the C comes right before the E?1/51/61/1205/6

Answer 1

To find the required probability, we need to find the number of ways in which C comes right before the E, and then divide that by the total number of sequences.

For the first, we have:

C E _ _ _ _

_ C E _ _ _

_ _ C E _ _

_ _ _ C E _

_ _ _ _ C E

For each of these cases, we can change the order of the other four letters. For example, for the first one, we can write:

C E A T I V

C E A T V I

C E A I T V

...

We can do that, for each of those 5 cases, a number given by:

4! = 4 * 3 * 2 * 1 = 24

Then, there are 5 * 24 = 120 ways to write C right before the E.

Now, the total number of sequences is given by:

6 * 5 * 4 * 3 * 2 * 1 = 720

This happens because we have 6 possible letters to pick first, then there will rest 5 possible letters to pick after that, and so on.

Finally, the probability we are looking for is:

P = 120/720 = 1/6

Therefore, the second option is correct.