Year after year, I have noticed the same thing - students are very conversant with solving P&C problems which have the condition "the four objects being together", but find it a tad difficult solving questions that have the condition "no two of the four objects are together". Just as there is a standard approach of handling "objects being together" - by considering all of them as one group, there is a standard approach to handle "no two objects being together". In this post, i will outline this standard approach and then take two questions that use this concept very innovatively.
In how many ways can four girls and five boys be arranged in a row such that no two girls are together?
The standard approach to ensure that no two girls are together is: Arrange the others first and then arrange the girls, with only one girl in the space between the others.
Thus, the 5 boys can be arranged in 5! ways. Now there are 6 positions in which the 4 girls can sit with only 1 girl in a position. This can be done in 6P4 ways or 6 x 5 x 4 x 3 ways. Thus the answer is 5! x 6 x 5 x 4 x 3
Now for the innovative question/solution based on the above...
1. In how many ways can 5 soldiers be selected from among 15 soldiers standing in a row such that no two of the selected soldiers are standing consecutively?
2. There are 15 empty cages arranged in a row. In how many ways can a lion, a tiger, a leopard, a panther and a puma be put in five of the cages with 1 animal in each such that no two occupied cages are consecutive?
Also spend a moment thinking if the two questions are the same or different.
Solutions tomorrow....
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