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 $7! \cdot (7C4 \cdot 5C3)$.
Best Answer
Your answer $7^4 \cdot 5^3$ is the number of ways the first four positions can be filled with consonants and the last three positions can be filled with vowels when letters may be repeated. However, the problem does not specify the positions of the consonants and vowels, so we must select them.
Notice that selecting which four of the seven positions will be filled with consonants also determines the positions of the three vowels since each of the remaining three positions must be filled with vowels.
We must select which four of the seven positions must be filled with consonants, which can be done in $\binom{7}{4}$ ways. Each of those four selected positions can be filled with a consonant in seven ways. Each of the remaining three positions can be filled with a vowel in five ways. Hence, there are $$\binom{7}{4}7^4 \cdot 5^3$$ words that can be formed consisting of four consonants and three vowels selected from an alphabet of seven consonants and three vowels with repetition.
The stated answer $7^7\dbinom{7}{4}\dbinom{5}{3}$ is nonsense.
Your answer $P(7, 4)P(5, 3)$ is the number of ways of filling the first four positions with distinct consonants and the last three positions with distinct vowels. However, the problem does not specify the positions of the consonants and vowels, so we must select them.
As above, we select four of the seven positions for the consonants. There are $P(7, 4)$ ways to arrange four of the seven consonants in those positions and $P(5, 3)$ ways to arrange three of the five vowels in the remaining three positions. Hence, there are $$\binom{7}{4}P(7, 4)P(5, 3)$$ words that can be formed consisting of four distinct consonants and three distinct vowels selected from an alphabet of seven consonants and five vowels.
Notice that $$\binom{7}{4}P(7, 4)P(5, 3) = \frac{7!}{4!3!} \cdot \frac{7!}{3!} \cdot \frac{5!}{2!} = 7! \cdot \frac{7!}{4!3!} \cdot \frac{5!}{2!3!} = 7!\binom{7}{4}\binom{5}{3}$$ The author of your book arrived at the answer by selecting four of the seven consonants in $\binom{7}{4}$ ways, selecting three of the five vowels in $\binom{5}{3}$ ways, and then arranging the seven distinct selected letters in $7!$ ways.