A falling number is an integer whose decimal representation has the property tha

Question

A falling number is an integer whose decimal representation has the property that each digit except the units digit is larger than the one to its right. For example 96521 is a falling number but 89642 is not. How many n-digit falling numbers are there, for n = 5 and 6? Suppose that A = {1,2,3, ...,10}. How many pairs of subsets of A like (X,F) are there such that: X = A and |Y|=3? X Y = theta and X Y = A? X Y = 2 and X Y = A? X r=theta and |x r|=8? In how many ways can n men and n women be seated around a table, alternating gender? A total of 6 freshmen, 5 sophomores and 4 juniors have volunteered to serve on a 4-person committee. How many such committees are possible if at least one freshen must serve on the committee?

Explanation / Answer

There are multiple questions to this part. Please post 1 more question for the same. I have answered the 3 parts.

15.1

for n=5 it will be 10c5

for n=6 it will be 10c6

Here the logic is that we need to choose unique numbers from the available 10 numbers.

15.3

n women can be seated in n! way and other men can be seated in (n-1)!way

so total will be n! x (n-1)!

15.4

atleast one freshmen should be there

so 6c1 * 9c3 + 6c2*9c2 + 6c3*9c1 + 6c4

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