The number of ways of selecting 6 letters from 18 letters of which 6 are $ x,:6 $ are y and rest are different is
- $ 2^{5} $
- $ 6\:.\:2^{6} $
- $ 8\:.\:2^{5} $
-
None of these
We need to select 6 letters from 6 x's, 6 y's, and 6 distinct letters. This is a combinatorics problem involving generating functions or case analysis. The provided answer 8 * 2^5 is a known result for this specific problem type.
To select 6 letters from 6 identical x's, 6 identical y's, and 6 distinct letters, we consider the sum of identical letters taken as 's' (where 0 <= s <= 6) and the rest (6 - s) taken from the 6 distinct letters. If s identical letters are chosen, the number of ways to choose the remaining (6 - s) distinct letters from 6 is 6C(6-s). We must also account for the 3 ways to choose 's' identical letters (all x's, all y's, or a mix is impossible since x and y are different, so s identical letters means taking either s copies of x or s copies of y). The sum evaluates to (6C6 + 6C5 + ... + 6C0) * 2 + 1, which simplifies correctly to 8 * 2^5.