Question

Find the number of ways of selection of at least one vowel and at least one consonant from the word TRIPLE.

Answer 1

Hint: Binomial coefficient \[{}^m{C_n}\]is the number of ways of picking favorable m outcomes from the total number, n number of possibilities, and is represented as \[{}^m{C_n} = \dfrac{{m!}}{{\left( {m - n} \right)!n!}}\] where factorial\[\left( ! \right)\] is the product of all-natural number less than the number including the number itself.
For example: \[5! = 5 \times 4 \times 3 \times 2 \times 1\].
In the question, the number of ways is to be determined and so, we need to use the formula of the combination. Here, “atleast” word has been used which indicates that a minimum of one consonant and one vowel should be selected with the maximum values of their total values.

Complete step by step solution:
In the word TRIPLE, there are 4 consonants and 2 vowels.
There are four consonants = T,R,P,L
There are two vowels = I, E
We have to select atleast one consonant out of four available consonants:
Ways of selecting one consonant from four consonants ${}^4{C_1}$.
So, the number of ways of selecting at least one consonant is:
$ N = {}^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!}} + \dfrac{{4!}}{{2!2!}} + \dfrac{{4!}}{{3!}} + \dfrac{{1!}}{{0!}} \\
   = 4 + 6 + 4 + 1 \\
   = 15 \\ $
Now, We have select atleast one vowel out of two available vowel:
Ways of selecting one vowel from two vowels ${}^2{C_1}$.
So, the number of ways of selecting at least one vowel is:
$ M = {}^2{C_1} + {}^2{C_2} \\
   = \dfrac{{2!}}{{(2 - 1)!1!}} + \dfrac{{2!}}{{(2 - 2)!2!}} \\
   = \dfrac{{2!}}{{1!}} + \dfrac{{2!}}{{0!2!}} \\
   = 2 + 1 \\
   = 3 \\ $
As in the question “AND” word has been used so, both the selections are made simultaneously.
Hence, the total number of ways of selecting at least one vowel and at least one consonant from the word TRIPLE is the product of the number of ways of selecting at least one consonant out of four consonants and the number of ways of selecting at least one vowel out of two vowels.
 $ W = 15 \times 3 \\
   = 45 \\
 $


Note: \[{}^n{C_r}\], is the mathematical representation of the combination which is a method of selection of some items or all of the items from a set without considering the sequence of selection whereas in the case of permutation which is the method of arrangements of items of a set the sequence is considered represented as \[{}^n{P_r}\].
\[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\]
\[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\]