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Four coffee mugs and a water bottle are to be chosen from eight different coffee mugs and four different water bottles and arranged in a row on the shelf with the water bottle in the center. What is the total number of such arrangements?

Option: 1

101600


Option: 2

202500


Option: 3

201600


Option: 4

210600

 

 


Answers (1)

best_answer

Given that,

There are 8 coffee mugs and 4 water bottles.

From 8 coffee mugs, the number of ways to choose 4 coffee mugs is given by,

\begin{aligned} &{ }^8 C_4=\frac{8 !}{4 !(8-4) !}\\ &{ }^8 C_4=\frac{8 !}{4 ! 4 !}\\ &{ }^8 C_4=70 \end{aligned}

From 4 water bottles, the number of ways to choose 1 water bottle is given by,

\begin{aligned} { }^4 C_1 & =\frac{4 !}{1 !(4-1) !} \\ { }^4 C_1 & =4 \end{aligned}

From the given, the water bottle should be placed in the center.

As a result, the remaining 6 coffee mugs can be organized in 6! different ways.

Thus, the required number of arrangements is given by,

\begin{aligned} &{ }^8 C_4 \times{ }^4 C_1 \times 6 !=70 \times 4 \times 6 !\\\\ &{ }^8 C_4 \times{ }^4 C_1 \times 6 !=201600 \end{aligned}

Therefore, the total number of arrangements is 201600 ways.

 

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