Question

55)

A) In how many dierent ways can the letters of the word 'JUDGE' be arranged such that the vowels

always come together?

B) How many 3 digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9 which are divisible by 5

and none of the digits is repeated?

C) In how many ways can 10 engineers and 4 doctors be seated at a round table without any restriction?

D) In how many ways can 10 engineers and 4 doctors be seated at a round table if all the 4 doctors sit

together?

Answer 1

A) The word JUDGE has 5 letters. It has 2 vowels [UE] and these 2 vowels should always come together. Hence these 2 vowels can be grouped and considered as a single letter. That is, JDG[UE].

Hence we can assume total letters as 4 and all these letters are different. Number of ways to arrange these letters

= 4!=4×3×2×1=24

In the 2 vowels [UE], all the vowels are different. Number of ways to arrange these vowels among themselves

=2!=2×1=2

Total number of ways =24×2=48

B) Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.

The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.

The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.

Required number of numbers = (1×5×4)(1×5×4) = 20.

C)  

Number of circular permutations (arrangements) of n distinct things
=(n−1)!

Initially let's find out the number of ways in which 10 engineers and 4 doctors can be seated at a round table.
In this case, n = total number of persons =10+4=14

Hence, number of ways in which 10 engineers and 4 doctors can be seated at a round table without any restriction
=(14−1)!=13!

D)

Number of circular permutations (arrangements) of n distinct things
=(n−1)!

Initially let's find out the number of ways in which 10 engineers and 4 doctors can be seated at a round table.
In this case, n = total number of persons =10+4=14=10+4=14

Hence, number of ways in which 10 engineers and 4 doctors can be seated at a round table
=(14−1)!=13!....(A)

Now let's find out the number of ways in which 10 engineers and 4 doctors can be seated at a round table where all the 4 doctors sit together.

Since all the 4 doctors sit together, group them together and consider as a single doctor.
Hence, n = total number of persons =10+1=11
These 11 persons can be seated at a round table in (11−1)!=10! ways ...(B)
The 4 doctors can be arranged among themselves in 4! ways ...(C)

From (B) and (C), number of ways in which 10 engineers and 4 doctors can be seated at a round table where all the 4 doctors sit together
=10!×4!