Question

Number of ways of selection of at least one vowel and one consonant from the word TRIPLE are
A.65
B.55
C.45
D.15

Answer 1

Hint: From the given word TRIPLE find the number of consonants and number of vowels. Given question has the condition of selection of at least one vowel and one consonant, so, find the number of ways of selection of at least one vowel by adding selection one vowel and selection of two vowels and so on till the number of vowels in the given word exceeds. Similarly, find the number of ways of selection of at least one consonant by adding selection of one consonant and selection of two consonants and so on till the number of consonants in the given word exceeds. Then by multiplying both the values we get our required answer.

Complete step by step answer:
The given word is TRIPLE, Now
The consonants in the word TRIPLE are T, R, P and L. The total number of consonants present in the given word TRIPLE is 4. Now, let us find the number of ways of selection of at least one consonant from the given 4 consonants.
\[{}^{4}{{C}_{1}}+{}^{4}{{C}_{2}}+{}^{4}{{C}_{3}}+{}^{4}{{C}_{4}}=\dfrac{4!}{(4-1)!1!}+\dfrac{4!}{(4-2)!2!}+\dfrac{4!}{(4-3)!3!}+\dfrac{4!}{(4-4)!4!}\]
\[=\dfrac{4.3.2.1}{3.2.1.1}+\dfrac{4.3.2.1}{2.1.2.1}+\dfrac{4.3.2.1}{1.3.2.1}+\dfrac{4.3.2.1}{4.3.2.1}=4+6+4+1=15\]
\[\therefore \]Number of ways of selection of at least one consonant from the given 4 consonants is 15.
The vowels in the word TRIPLE are I and E. The total number of vowels present in the given word TRIPLE is 2. Now, let us find the number of ways of selection of at least one vowel from the given 2 vowels.
\[{}^{2}{{C}_{1}}+{}^{2}{{C}_{2}}=\dfrac{2!}{(2-1)!1!}+\dfrac{2!}{(2-2)!2!}=\dfrac{2}{1.1}+\dfrac{2}{2}=2+1=3\]
\[\therefore \]Number of ways of selection of at least one vowel from the given two vowels is 3.
Now to find the number of ways of selection of at least one vowel and one consonant from the word TRIPLE we have to multiply the values we got
\[\Rightarrow \] (number of ways of selection of at least one vowel) \[\times \](number of ways of selection of at least one consonant)
\[=15\times 3\]
\[=45\]
\[\therefore \] Total number of ways of selection of at least one vowel and one consonant from the word TRIPLE is 45.
\[\therefore \] Option C is correct.
Note:
The possibilities for making mistakes in this type of problems are, one may make a mistake by considering the number of ways of selection of at least one vowel and one consonant as the number of ways of selection of one vowel and one consonant.