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4 in how many ways can 7 children be seated in a

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4. In how many ways can 7 children be seated in a round table?
5. In how many ways can a family of 6 members be seated in a circular tableif two of them insist sitting beside each other?If two of them insist to sit beside each other, count them as one.
6. Mario bought 4 different Mathematics books (Algebra, Geometry, Statisticsand Trigonometry) and 5 different Science books (Biology, Chemistry,Physics, Natural Science, and Anatomy). If he wants to put them on a shelf,how many ways can Mario arrange these books on a shelf ifa.no restrictions?b.books of the same subject must be placed together?
Solution:
6MathAlgebraGeometryStatisticsTrigonometryScience Biology, Chemistry, Physics Natural, Science, Anatomysince Math and Science can interchange their positions, so it’s 2!4! · 5! · 2! = 4·3·2·1·5·4·3·2·1·2·1 =5, 760ways7.Nena’s house is three blocks north and five blocks west from her school.How many routes can Nena take from her house to school if she alwayswalks in either south or east?HouseNSschoolConsider some various ways that Nena can walk from her house toschool in either south or east.WE
7Notice that in each of the possible routes, the total number of blocks is8. Also notice that some elements that are identical to each other, E showsup 5 times and S shows up 3 times. Thus, we must treat this as a permutationproblem in which some of the elements are identical. Keeping that in mind,the number of routes she can walk would be:P=8!5!3!=8·7·6·5!5!3!=8·7·63·2·1= 8·7 =56By walking, there are 56 routes from her house to school in either south oreast direction.

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