how many words of $4$ consonants and $3$ vowels can be made from $12$ consonants and $4$ vowels? if all the letters are different?

Question

my effort for this question is

I am selecting $4$ consonants from $12$ available consonants and $3$ from $4$ available vowels. After selecting $4$ consonants and $3$ vowels now I have $7$ letters in my hand now I am permuting them all with $7!$. So the total number of words can be made is ${12\choose4}{4\choose3}7!$.

But the answer is ${12\choose4}{4\choose3}$.

I would appreciate if anyone advises on this question.

Answer 1

I think you are correct

To take a simpler example, with two consonants from three $\{R,S,T\}$ and one vowel from two $\{A,I\}$, there are $3C2 \cdot 2C1 \cdot 3! = 36$ possibilities namely $$RSA, RSI,STA,STI,TRA,TRI,SRA,SRI,TSA,TSI,RTA,RTI,$$ $$RAS, RIS,SAT,SIT,TAR,TIR,SAR,SIR,TAS,TIS,RAT,RIT,$$ $$ARS, IRS,AST,IST,ATR,ITR,ASR,ISR,TSA,ITS,ART,IRT$$

not $3P2 \cdot 2P1 = 12$, which would just be the first row in my example of having all the consonants first and then all the vowels