E-Storepermutations and combinations problems.
Some of the following problems will take some time to solve. Don’t get frustrated. Enjoy them!!
Q1) in how many ways can we arrange the numbers 21 , 31 , 41, 51, 61 ,71 and 81 such that any 4 consecutive numbers is divisible by 3? [answer : 144]
Q2) for how many pairs of consecutive integers in {1000, 1001, 1002 …. , 2000}
Is no carrying required when we add them? [answer 156]
Q3) determine the number of ways to choose 5 numbers from the first 18 positive integers such that any 2 of them differ by at least 2. [answer: 2002]
Q4) 25 boys and 25 girls are seated around a round table. Show that it is always possible to find a person in any arrangement possible both of whose neighbors are girls.
Q5) prove that among any 16 positive integers from 1 to 100 where all of them are distinct, it is always possible to 4 integers a, b, c and d such that a + b = c + d. [think pigeon hole principal!!!]
Q6) suppose we have a 7*7 checker – board. We paint any 2 of its squares with the colour yellow and the rest are painted green. Two colour schemes are equivalent if one can be obtained from the other by rotating the board. Find the number of inequivalent colour schemes. [answer: 300]
Q7) a set of positive integers has the triangle property if it has three distinct elements which are lengths of the sides of a triangle whose area is positive.
Consider the sets {4, 5, 6, ….. n} of consecutive positive integers all of whose 10 element subsets have the triangle property. What is the largest possible value of n? [answer: 253]
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