To count the number of different words consisting of the letters a,b,c,d,e,p that meet the given conditions, we can use the multiplication principle of counting.
First, we need to choose the positions for the letters c and e. Since we want the word to include both c and e, we have two choices for the position of c and one choice for the position of e, or vice versa. This gives us a total of 4 different ways to choose the positions for c and e.
Next, we need to fill in the remaining two positions with any of the remaining letters: a, b, d, or p. There are 4 choices for the first remaining position, and 3 choices for the second remaining position, since we cannot use the same letter twice.
Therefore, the total number of different words that meet the given conditions is:
4 (choices for the positions of c and e) x 4 (choices for the first remaining position) x 3 (choices for the second remaining position) = 48
So there are 48 different words consisting of the letters a,b,c,d,e,p that do not start with the letter p and do not end with the letter b and include the letters c,e.