The letters A,B,C,D,E,F, and G are arranged in random order.

Question

The letters A,B,C,D,E,F, and G are arranged in random order.

(a) What is the probability that A and E are next to each other?

(b) What is the probability that A, B, and C appearing in a row in some

order in the string of letters?

Answer 1

Let's say a and e can be placed in the first and second position. That means we have 2*1*5*4*3*2*1 selections. The total possibilities are 7! When you divide these 2 numbers and cancel a lot of the numerator with the denominator, you get 2/42 or 1/21. Now the ae combination can be in second and third positions, 3rd and 4th, 4th and 5th, 5th and 6th and 6th and 7th. So multiply 6*(1/21) to get 2/7.

 

abc_ _ _ _ would be 3*2*1*4*3*2*1 divided by 7! is 1/35. multiply that by 5 (1st, 2nd, 3rd,  2nd,3rd,4th   3rd 4th 5th   4th, 5th 6th    5th, 6th 7th ) to get 1/7.