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  • A string of 10 lights needs to be arranged, where 5 are red, 3 are green, and 2 are blue. How many different arrangements are possible?  <div class='qn

A string of 10 lights needs to be arranged, where 5 are red, 3 are green, and 2 are blue. How many different arrangements are possible?

 

Option: 1

10,080

 


Option: 2

15,500

 


Option: 3

12,250

 


Option: 4

20,850


Answers (1)

best_answer

To calculate the number of different arrangements of the 10 lights, where 5 are red, 3 are green, and 2 are blue, we can use the concept of permutations.

First, let's consider the total number of arrangements without any restrictions. Since there are 10 lights in total, they can be arranged on the string in 10 ! (10 factorial) ways.

However, within these arrangements, the lights of the same color are identical. So, we need to account for the repetition of the colors.

The number of ways to arrange the 5 red lights among themselves is given by 5 !, and the number of ways to arrange the 3 green lights among themselves is given by 3 !. Similarly, the number of ways to arrange the 2 blue lights among themselves is given by 2 !.

To find the total number of different arrangements with the given color distribution, we divide the total number of arrangements by the number of ways the lights of the same color can be arranged:

\mathrm{ 10 ! /(5 ! \times 3 ! \times 2 !) . }

Calculating this expression, we get:

\mathrm{ 10 ! /(5 ! \times 3 ! \times 2 !)=(10 \times 9 \times 8 \times 7 \times 6) /(3 \times 2)=30 \times 8 \times 7 \times 6=30 \times 336=10,080 . }

Therefore, there are 10,080 different arrangements possible for the string of 10 lights, where 5 are red, 3 are green, and 2 are blue.

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seema garhwal

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