Visual SolutionPYQ 2023 · Apr 8 Shift 1Tricky

Question

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

(A)

16800

(B)

33600

(C)

18000

(D)

14800

Solution Path

Vowels as one block:

\dfrac{8!}{3! \cdot 2!} \times \dfrac{5!}{4!} = 3360 \times 5 = 16800

01Question Setup

1/4

Arrangements of INDEPENDENCE with all vowels together.

\text{Ways} = \;?

02Identify Letters

2/4

Vowels: I, E, E, E, E (5). Consonants: N, D, P, N, D, N, C (7). Treat vowels as one unit: 8 objects total.

8 objects = 7 consonants + 1 vowel block

03Count ArrangementsKEY INSIGHT

3/4

Arrange 8 objects (3 N's, 2 D's):

\dfrac{8!}{3! \cdot 2!} = 3360

. Vowels within block:

\dfrac{5!}{4!} = 5

. Total:

3360 \times 5 = 16800

3360 \times 5 = 16800

04Final Answer

4/4

Number of arrangements

= 16800

\boxed{16800}

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