Question

How many 8-letter permutations of 10 letters A,D,E,G,H,K,O,T,W,Z (i) have at least one of the subwords DOG, ATE, HW? (ii) have none of the subwords DOG, ATE, HW? (iii) if the letters are lined up randomly what is the probability to spell at least one of those subwords?

Answer 1

To find the number of 8-letter permutations of 10 letters with specific subwords, we can use the principle of inclusion-exclusion. The probability of spelling at least one of those subwords can be found by dividing the number of permutations with at least one subword by the total number of permutations.

Step-by-step explanation:

To answer this question, we can use the principle of inclusion-exclusion. We'll start by finding the total number of 8-letter permutations using the 10 given letters, which is 10P8 = 10!/2! = 34,560.

(i) To find the number of permutations that have at least one of the subwords DOG, ATE, or HW, we'll individually count the permutations with each subword and then subtract the overlapping permutations.

(ii) To find the number of permutations that have none of the subwords DOG, ATE, or HW, we'll subtract the number of permutations with at least one of those subwords from the total number of permutations.

(iii) To find the probability of spelling at least one of those subwords when the letters are lined up randomly, we'll divide the number of permutations with at least one of the subwords by the total number of permutations.