First slide

Combinations

Question

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is

Moderate

A

less than 500
B

at least 500 but less than 750
C

at least 750 but less than 1000
D

at least 1000

Solution

Four novels can be selected from 6 novels in $^{6} C_{4}$ ways. One dictionary can be selected from 3 dictionaries in $^{3} C_{1}$ ways.
As the dictionary selected is fixed in the middle, the remaining 4 novels can be arranged in 4! ways.
$∴$ The required number of ways of arrangement $=^{6} C_{4} \times^{3} C_{1} \times 4! = 1080$

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