Question

How many words can be made out from the letters of the word INDEPENDENCE, in which vowels always come together
A.\[16800\]
B.\[16630\]
C.\[1663200\]
D.None of these

Answer 1

Hint: First we know a permutation is an act of arranging the objects or numbers in order. Consider each option, we find the solution using the formula, total number words formed by the given word\[ = \dfrac{{n!}}{{{r_1}!\; \times {r_2}!\; \times - - - \times {r_t}!}}\]Where \[n\]be the number of letters in the given word and \[{r_1},{r_2},....,{r_t}\]be the multiplicity of the letters which are repeated in the given word.

Complete step-by-step answer:
The factorial of a natural number is a number multiplied by "number minus one", then by "number minus two", and so on till \[1\] i.e., \[n! = n \times \left( {n - 1} \right) \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times - - - \times 2 \times 1\] . The factorial of \[n\] is denoted as \[n!\]. \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!\;\;}}\] where, \[n\] is the total items in the set and \[r\]is the number of items taken for the permutation find the value of \[n\].
Given the word “INDEPENDENCE”.
Since vowels are “a or A”, “e or E”, “ i or I”, ”o or O”, “u or U”.
Hence the given word “INDEPENDENCE” contains 5 vowels
Then “IEEEE” are taken together, we may treat them as one letter. Then, the word can be written as “NDPNDNC(IEEEE)” and number of letters in the word is \[8\]. Since “N” letter repeated three times and the letter “D” is repeated 2 times.
Then the number of words formed by \[8\] letters of the word is \[\dfrac{{8!}}{{3! \times 2!}} = 3360\].
Since in the word “IEEEE” the letter “E” is repeated 4 times
Then the number of words formed by the word IEEEE is \[\dfrac{{5!}}{{4!}} = 5\].
Hence, the total number of words can be made out from the letters of the word “INDEPENDENCE” in which vowels always come together \[ = 5 \times 3360 = 16800\].
\[ \Rightarrow \] The option (a) is correct.
So, the correct answer is “Option A”.

Note: Note that Combinations are the way of selecting the objects or numbers from a group of objects or collection, in such a way that the order of the objects does not matter. Using the combination formula \[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!\;\;r!\;}}\] where, \[n\] is the total items in the set and \[r\]is the number of items taken for the permutation to find the value of \[n\].