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In how many ways can the letters of the word ‘THERAPY’ be arranged so that the vowels never come together ?
720
1140
5040
3600
4800
720
1140
5040
3600
4800
Solution
SOLUTION 1:
The word THERAPY consists of 7 distinct letters with two vowels A and E.
The 5 consonants can be arranged in 5! ways and the remaining 2 vowels can be arranged in the gaps so that the vowels do not come together.
_ T_H_R_P_Y_
There are 6 gaps in between the consonants and the two vowels can be arranged in 6 gaps in 6P2 ways.
Therefore,
Total ways in which the letters of the word ‘THERAPY’ be arranged so that the vowels never come together is
= 5! × 6P2
= 120 × 30
= 3600
SOLUTION 2:
7 letters can be arranged in 7! ways.
7! = 5040
Consider a case where the two vowels always come together. (Two vowels are grouped)
When the two vowels are grouped then the total number of letters is
= 5 consonants and 1 group = 6
The 6 letters can be arranged in 6! ways.
6! = 720
The two vowels in the group can be arranged in 2! ways.
2! = 2
The number of arrangements in which the two vowels always come together is
= 720 × 2 = 1440
The number of arrangements in which the two vowels never come together is
= 5040 − 1440
= 3600
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Simple equations are mathematical expressions that involve one or more variables and follow the basic algebraic form. In these equations, the goal is to find the values of the variables that satisfy the given conditions. Simple equations typically consist of addition, subtraction, multiplication, and division operations, making them fundamental to understanding algebraic concepts.
