Quantitative Reasoning

Permutation & Combination

Medium

In how many different ways, can the letters of the word ASSOCIATION be arranged, so that the vowels always come together?

13456

63845

36780

32400

✅ Correct Option: 4

The word ASSOCIATION has 11 letters.

Vowels: A, A, O, O, I, I (6 vowels total)

Consonants: S, S, C, T, N (5 consonants total)

Treat all vowels as one single block since they must stay together.

Items to arrange:

1 vowel block (AAOOII)
5 consonants (S, S, C, T, N)

Total items = 6

Arranging these 6 items where S repeats 2 times:

Number of arrangements $= \dfrac{6!}{2!}$

$= \dfrac{720}{2}$

$= 360$

Arranging the vowels within their block:

The vowel block contains: A, A, O, O, I, I (6 vowels)

Where A repeats 2 times, O repeats 2 times, and I repeats 2 times:

Number of arrangements $= \dfrac{6!}{2! \times 2! \times 2!}$

$= \dfrac{720}{2 \times 2 \times 2}$

$= \dfrac{720}{8}$

$= 90$

Total arrangements $= 360 \times 90$

$= 32,400$

2026: 30 May Shift 1

How many three digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digit is repeated?

2026: 20 May Shift 2

How many 5-digit numbers can be formed by using digits 1, 2, 3, 4, 5 if 2 and 3 cannot come together?

2025: 21 May Shift 1

In how many ways are the letters of the word DOCUMENT arranged so that all the vowels always come together?

2026: 18 May Shift 2

Match the LIST-I with LIST-II

LIST-I (Expression/Statements etc.)	LIST-II (Values etc.)
A. $^5P_5 - ^{10}C_0$	I. 22
B. $^{15}C_{11} - ^{16}C_2$	II. 2520
C. If $^nP_3 = 9240$ , then find the value of n.	III. 0
D. In how many ways, the letters of the word 'BANKING' be arranged?	IV. 1245

Choose the correct answer from the options given below:

2025: 15 May Shift 1

Read the information given below carefully and answer the question that follows:

(A) The different ways in which the alphabets of the word BAKERY can be arranged is 720

(B) The number of ways in which the alphabets of the word MACHINE can be arranged so that the vowels will occupy only the odd positions is 576

Choose the correct answer from the options given below:

2025: 3 June Shift 1

How many ways can 10 persons shake hands with two persons?

In how many different ways, can the letters of the word ASSOCIATION be arranged, so that the vowels always come together?

Solution

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