Question

Let E denote the set of letters of the English alphabet, \[\;V = \left\{ {a,e,i,o,u} \right\}\] and C be the complement of V in E. Then, the number of four letter words (where repetition of letters are allowed) having at least one letter from V and at least one letter form C is:
A) \[261870\]
B) \[314160\]
C) \[425880\]
D) \[851760\]

Answer 1

Hint: To solve this problem, we will first find the set of E, now since the set of V is already given to us, so, with the help of E and V, we will find the set of C, as C is the complement of V in E. Now using all the values, we will find the total number of four-letter words with repetition, and hence with that, we will get our required answer.

Complete step-by-step answer:
We have been given that E denotes the set of letters of the English alphabet, \[V = \left\{ {a,e,i,o,u} \right\}\] and C be the complement of V in E. We need to find the number of four letter words, where repetition of letters is allowed, having at least one letter from V and at least one letter from C.
So, it is given that E is the set of letters of the English alphabet, and there are \[26\] letters in the English alphabet.
Also, it is given that, V is the set of vowels, and there are \[5\] vowels in the English alphabet i.e., \[\left\{ {a,e,i,o,u} \right\}.\]
And, C is the complement of V in E, i.e., \[C = E - V \Rightarrow C = 26 - 5 = 21,\] i.e., set of \[21\] letters from the English alphabet.
Now, we have to form four letter words, having at least one letter from V and at least one letter form C.
Therefore, total words with repetition\[ = 26 \times 26 \times 26 \times 26 - 5 \times 5 \times 5 \times 5 - 21 \times 21 \times 21 \times 21\]
\[
= {(26)^4} - {(5)^4} - {(21)^4} \\
= 456976 - 625 - 194481 \\
= 261870 \\
 \]
So, the total number of four-letter words with repetition are \[261870.\]
So, the correct answer is “Option A”.

Note: Here, complement word is present in the question, let us understand about it in detail. We know that the probability of all possible events is equal to one, and for two events to be complements, that out of two events, one or the other must occur. Therefore, the probabilities of an event and its complement should always be equals to one.