From $7$ consonants and $5$ vowels, how many words can be formed consisting of $4$ consonants and $3$ vowels if:
- Any letter can be repeated.
- No letter can be repeated.
My answer for the 1st part was $7 \cdot 7 \cdot 7 \cdot 7 \cdot 5 \cdot 5 \cdot 5$ (or $7^4 \cdot 5^3$). The way I thought about it is there are $4$ available slots for $7$ consonants that can be repeated so despite taking a consonant which we call it $c_1$, the second/third/fourth slot might also be $c_1$ so each of the $4$ slots has $7$ consonants to choose from so its $7 \cdot 7 \cdot 7 \cdot 7$. Same case for vowels which is $5 \cdot 5 \cdot 5$. Multiply those together to get total number of ways where the letter is repeated in a word which is $7^4 \cdot 5^3$.
However it turned out that the correct answer is actually $7^7 (7C4 \cdot 5C3)$. I don't understand the logic behind this. I don't get why not only the combinations nCr is used which is supposed to be used for the cases that don't require order and repetition, but also it is multiplied by $7^7$.
For the 2nd part, since the letter cannot be repeated then the available consonants and vowels decreases every time you choose from them so its $7 \cdot 6 \cdot 5 \cdot 4 \cdot 5 \cdot 4 \cdot 3$ (or $7P4 \cdot 5P3$).
However that's not the case because the correct answer is $7! \cdot (7C4 \cdot 5C3)$.