If a code word is defined to be a sequence of different letters chosen from the 10 letters A, B, C, D, E, F, G, H, I, and J, what is the ratio of the number of 5-letter code words to the number of 4-letter code words?

The answer is 6:1 .

Explanation:

We don't need to have prior knowledge of combination or permutation.

Count the way to chose 4 letters.As given in the question chose sequence of different letters

For first place we have 10 letter

For second place we have 9 letter as one is already out

For 3 place we have 8 letter as two letter are already out

For 4 place we have 7 letter as three letter are already out

Total group of 4 letters = 10 * 9 *8*7

Similarly form group of 5 letters we have

Total group of 5 letters = 10 * 9* 8 * 7*6

then Calculating ratio between 4 and 5

We have,

Group 5 : Group 4 ::10 * 9* 8 * 7*6:10 * 9 *8*7

                              : : 30240 : 5040

divide by 5040 on both side we get

Group  5: Group 4 :: 6 : 1