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Question: The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 ...

The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?

Explanation

Solution

Before attempting this question, one should have prior knowledge about the concept of permutation and combination and also remember to use the formula nCr=n!(n−r)!r!^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}and nPr=n!(n−r)!^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}, use this information to approach the solution.

Complete step-by-step answer:
According to the given information we have to choose 2 different vowels from 5 vowels and 2 different from 21 constants
So, to choose 2 vowels from 5 vowels the number of ways is 5C2^5{C_2}
We know that formula of combination is given as nCr=n!(n−r)!r!^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}here n is the total numbers of objects in the given set and r is the number of objects we have to choose from the set
Therefore, 5C2=5!(5−2)!2!=5×42^5{C_2} = \dfrac{{5!}}{{\left( {5 - 2} \right)!2!}} = \dfrac{{5 \times 4}}{2}
5C2^5{C_2}= 10 ways
Also, we have to choose 2 different consonants from 21 constants so the number of possible ways is 21C2^{21}{C_2}
Therefore, 21C2=21!(21−2)!2!=21×202^{21}{C_2} = \dfrac{{21!}}{{\left( {21 - 2} \right)!2!}} = \dfrac{{21 \times 20}}{2}
21C2^{21}{C_2}= 210 ways
So, the number of possible arrangements of 4 letters is 4P4=4!(4−4)!=4×3×2×1=24^4{P_4} = \dfrac{{4!}}{{\left( {4 - 4} \right)!}} = 4 \times 3 \times 2 \times 1 = 24
So, the total number of ways to select the 4 letters = 24 ways

Thus, the total numbers of words that can be formed is equal to the number of ways selecting 2 vowels ×\times number of ways selecting 2 consonants ×\times number of arrangements of 4 letters
Therefore, the total number of words of 4 letters = 21C2×5C2×4!=210×10×24=50400^{21}{C_2}{ \times ^5}{C_2} \times 4! = 210 \times 10 \times 24 = 50400
Therefore, the total number of words of 4 letters is 50400 words.

Note: In the above solution we used the method of permutation and combination to get the required result where permutation can be explained as arrangement of an element from the set into a sequence whereas the combination defines the ways of choosing the elements from the set.