PERMUTATIONS & COMBINATIONS–01

1. Question: In how many different ways can the letters of the word PENCIL be arranged?  [Ans. (c) 720]

(a) 120     (b) 240    (c) 720   (d) 719    (e) None of these

2. Question: In how many different ways can the letters of the word JUDGE be arranged so that the vowels always come together? [Ans. (a) 48]

(a) 48     (b) 24    (c) 120   (d) 60    (e) None of these

3. Question: Letters of the word DIRECTOR are arranged in such a way that all the vowels come together. Find out the total no. of ways for making such arrangement? [Ans. (d) 2160]

(a) 4320   (b) 2720   (c) 1120   (d) 2160   (e) None of these

Solution:

Detailed Explanation:

1. Solution: In the word “PENCIL”, there are six different letters, that is, six different things, that can be arranged by 6! Ways.

6! = 6×5×4×3×2×1 = 720. Ans.

2. Solution: In the given word “JUDGE”, since vowels should come together, so we take (UE) as one object and J,D and G as three other objects. So, we have total four different objects, which can be arranged by 4! Ways =4×3×2×1 = 24 ways. But, two letters U and E in the group (UE) can be arranged themselves by 2! Ways, that is , by 2×1 = 2 ways.

Therefore, total no. of ways = 24×2 = 48 ways. Ans.  

3. Solution: Similarly as in the question no. 2, in the given word “DIRECTOR”, since vowels should come together, so we take (IEO) as one object and D, C,T R and R as five other objects. So, we have total six objects, which can be arranged by 6! Ways =6 ×5 ×4×3×2×1 = 720 ways. But, three letters I, E and O in the group (IEO) can be arranged themselves by 3! Ways, that is , by 3 ×2×1 = 6 ways.  

Therefore, total no. of ways = 720×6 = 4320 ways. Also, the six objects above mention are not different. There are two R’s, so the correct answer is (4320/2) =2160 Ans.