3 consonants BCD.
3 vowels AEI.
Choose 2 consonants, no repetition.
Choose 2 vowels, no repetition.
Form different words from the chosen consonants and vowels.
Choose 2 consonants, 3 ways - BC, BD, CD.
Choose 2 vowels, 3 ways - AE, AI, EI.
Different groups with 2 consonants and 2 vowels, 3x3 = 9.
BCAE, BCAI, BCEI, BDAI, BDAI, BDEI, CDAE, CDAI, CDEI.
Different words from 4 character set such as BCAE, 4! = 4x3x2x1 = 24.
BCAE, BCEA, BACE, BAEC, BECA, BEAC.
CAEB, CABE, CEAB, CEBA, CBAE, CBEA.
ABCE, ABEC, AEBC, AECB, ACBE, ACEB.
EBCA, EBAC, ECBA, ECAB, EABC, EACB.
Total 4 character words from 2 different consonants and 2 different vowels would be 9x24 = 216.
Apply to above.
7C3 x 4C2 x 5! = 25200.
More difficult would be when repetitions are allowed in consonants and vowels.
Hint: Here, we are going to find the number of ways to choose vowel and consonant from the given word. Also, we are going to use a combination formula to solve the problem to get the required answer.Formula used:
Here, we were using the Combination formula,
\[{}^{\text{n}}{C_{\text{r}}}{\text{ = }}\dfrac{{{\text{n}}!}}{{{\text{[n - r]!}}}}\]
By using this formula, we will get the required answer.
Complete step by step answer:
It is given that the word, in which we need to find the number of ways to choose vowel and consonant, was ‘ALLAHABAD’
In the letter of the word ‘ALLAHABAD’ there is only one vowel available for selection that is [A].
Remember that the fact that A is available \[{\text{4}}\] times has no impact on this fact.
Also, there are \[{\text{4}}\] consonants available in the given word ‘ALLAHABAD’
“L, H, B and D” are the four consonants available in the word that has been given
Thus, the number of ways of selecting a vowel and a consonant would be,
\[ \Rightarrow \] Vowels can be chosen in \[{}^1{{\text{c}}_1}\] \[{\text{ = 1}}\] way.
\[ \Rightarrow \] Consonants can be chosen in \[{}^4{{\text{c}}_1}\] \[{\text{ = 4}}\] ways.
\[ \Rightarrow \] Total \[{{ = 1 \times 4 = 4}}\]
Hence, the number of ways to choose vowels and consonants is \[{\text{4}}\].
Note:
There is an alternative method, which can be used to solve the problem.
In the word ‘ALLAHABAD’, we have four A’s and that only one vowel in the given word and
There are Four consonants in the given word. They are B, D, H, L
So, any four combinations:
AB, AD, AH, AL
So, there are only four ways.
Thus, the number of ways of selecting a vowel and a consonant is \[{\text{4}}\].