S5 maths permutation problem

Helon

2014-01-06 6:52 am

Four letters are taken from the word 'SECONDARY' and arranged in a row. Find
the number of ways of arranging them if no vowels are placed together.

I find it difficult to divide the cases, though I am sure 3 vowels cannot be taken.
Thanks!

回答 (1)

用戶2053

2014-01-06 8:12 am

✔ 最佳答案

Case 1 (No vowel) :
Any 4 from S,C,N,D,R,Y , 6P4 = 360 ways.
Case 2 (1 vowel) :
Any 3 from S,C,N,D,R,Y and 1 from E ,O ,A , 6C3 × 3C1 × 4! = 1440 ways.Alternatively : 6P3 × 3C1 × 4P1 = 1440 ways.
An example for explanation : ( ) S ( ) C (E) N ( )
Case 3 (2 vowels) :
Any 2 from S,C,N,D,R,Y and 2 from E ,O ,A ,
6C2 × 3C2 = 45 waysAll ways = 45 × 4! = 1080 ways.
2 vowels are placed together ways
= 45 × 2(arranging 2 vowels ways) × 3! = 540 ways
No vowels are placed together ways = 1080 - 540 = 540 ways.Alternatively : 6P2 × 3C2 × 3P2 = 540 ways.
An example for explanation : (A) S ( ) C (E)
Case 1 + Case 2 + Case 3 = 360 + 1440 + 540 = 2340 ways.

👍 0 👎 0