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CAT 2025 Lesson : Permutations & Combinations - Slotting Technique
2.3 Slotting Techniques
In these questions, we would be given two or more groups, say for example Groups A and B have and members respectively. The question might require us to find out the cases where members of Group B are not next to each other.For this, we need to first seat (permute) the members of Group A which is ways. We will then have slots (between the members and the two ends) where we can slot the members of group B, so that they are never adjacent to each other. Here we can select the slots and then arrange them in ways.
Example 16
boys and girls are to sit in a row. None of the boys should have another boy seated next to him. In how many ways can the be seated?
() () () ()
We now have slots between these students and at the ends as shown below (where G represents a girl seated and __ represents an empty slot). We can permute the boys in these slots in ways.
__ G __ G __ G __ G __ G __ G __ G __ G __
Total permutations =
Answer: ()
() () () ()
Solution
Permutation for seating the girls = waysWe now have slots between these students and at the ends as shown below (where G represents a girl seated and __ represents an empty slot). We can permute the boys in these slots in ways.
__ G __ G __ G __ G __ G __ G __ G __ G __
Total permutations =
Answer: ()
Example 17
In how many ways can girls and boys be seated such that
(I) none of the boys sit next to each other?
(II) none of the boys sit next to each other and none of the girls sit next to each other?
Total permutations = = =
Case II: Since neither the boys nor the girls sit next to each other, they sit in alternate positions. So, the seating from left, either starts with a girl (GBGBGBGB) or starts with a boy (BGBGBGBG). We have such arrangements.
We can seat the boys in the positions in ways and the girls in their respective positions in ways.
Total permutations = =
Answer: (I) ; (II)
(I) none of the boys sit next to each other?
(II) none of the boys sit next to each other and none of the girls sit next to each other?
Solution
Case I: We first place the girls in ways. We now have = slots to permute the boys in.Total permutations = = =
Case II: Since neither the boys nor the girls sit next to each other, they sit in alternate positions. So, the seating from left, either starts with a girl (GBGBGBGB) or starts with a boy (BGBGBGBG). We have such arrangements.
We can seat the boys in the positions in ways and the girls in their respective positions in ways.
Total permutations = =
Answer: (I) ; (II)
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