First slide

Theorems of probability

Question

If the letters of the word 'REGULATIONS' are arranged at random, then the probability that there will be exactly 4 letters between R and E is

Moderate

A

$\frac{11!}{^{9} C_{4} 6!}$
B

$\frac{12!}{^{9} C_{5} 6! 2!}$
C

$\frac{11!}{^{9} P_{4} 6! 2!}$
D

$\frac{11!}{^{9} C_{4} 6! 2!}$

Solution

Total number of ways of arranging 11 letters = (11)! ---------- (1)
Selection of 4letters to be placed between R and E from the remaining 9 $=^{9} C_{4}$
These 4letters can be permuted in 4! ways. Now permutation of
$R and E can be interchanged in 2! ways.$
$∴ Number of favourable ways =^{9} C_{4} 6! 2! - - - - - - (2)$
$Hence, required probability = \frac{11!}{^{9} C_{4} 6! 2!}$

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