Question

How many words can be formed with the letters of the word $\text{PATALIPUTRA}$ without changing the relative positions of vowels and consonants?
A. 720
B. 1800
C. 3600
D. 4320

Answer 1

Hint: Here, count the number of letters in the given word and also count the number of times a particular letter is repeated, and apply permutation formula to find the number of arrangements.

Complete step by step answer:
We are given the word “PATALIPUTRA” in which total number of letters are 11 (i.e. $P, A, T, A, L, I, P, U, T, R, A$).
Total number of vowels = $5 (i.e. 3 A’s, 1 I $ and $1 U)$
Total number of consonants = $6 (i.e. 2 P’s, 2 T’s, 1 L $ and $1 R)$
Also, given that words can be formed with the letters of the word PATALIPUTRA without changing the relative positions of vowels and consonants.
So, total number of words = $\dfrac{{5!}}{{3!}} \times \dfrac{{6!}}{{2!2!}}$
[Permutation formula: In arrangement of n letters in which letters a and b are repeated $x$ and $y$ times, then the number of possible arrangements is given as $\dfrac{{n!}}{{x!y!}}$.
Here, $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$
$5! = 5 \times 4 \times 3 \times 2 \times 1$
$3! = 3 \times 2 \times 1$
$2! = 2 \times 1$
Putting all values and simplifying, we get
Total number of words = $4 \times 5 \times 180 = 3600$

Therefore, the number of words that can be formed with the letters of the word $\text{PATALIPUTRA}$ without changing the relative positions of vowels and consonants is 3600. Hence, option (C) is correct.

Note:
In these types of question, first check whether question is asked about combination or
permutation. Permutation means arrangement of things, and combination means taking a particular number of items at a time (arrangement does not matter in combination). Then apply the proper formula as required. Observe that the given condition is with or without repetition condition and proceed for error free calculations.