JEE MAIN - Mathematics (2023 - 8th April Morning Shift - No. 1)
The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is :
16800
14800
18000
33600
Explanation
In the given word,
vowels are : I, E, E, E, E
Consonants are : N, D, P, N, D, N, C
So, number of words $=\frac{8 !}{3 ! 2 !} \times \frac{5 !}{4 !}$
$=\frac{8 \times 7 \times 6 \times 5 \times 4}{2} \times 5=16800$
Concept :
Out of $n$ objects, if $r$ things are same, so number of ways $=\frac{n !}{r !}$
vowels are : I, E, E, E, E
Consonants are : N, D, P, N, D, N, C
So, number of words $=\frac{8 !}{3 ! 2 !} \times \frac{5 !}{4 !}$
$=\frac{8 \times 7 \times 6 \times 5 \times 4}{2} \times 5=16800$
Concept :
Out of $n$ objects, if $r$ things are same, so number of ways $=\frac{n !}{r !}$
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