practice problems pending

Q1. Let X = {1,2,...,n}). In how many ways can you choose (r) elements from (X) such that no two chosen elements are consecutive, where 0 <= r <= (n+1)/2.
S1.
Answer: (n-r+1)c(r)
Solution here.

Q2. In how many ways can three numbers in arithmetic progression (A.P.) be selected from the set (1,2...n)?
S2.
a,b,c in A.P. => a,c are both even or both odd.
b = (a+c)/2

So if we pick 2 even or 2 odd numbers, we will find a number between them to complete the A.P.

Case 1:
n is even.
n = 2m.
m odd and m even numbers.
Pick 2 even: mC2
Pick 2 odd: mC2
Total: 2 * mC2 = m(m-1) = n/2(n/2-1) = n(n-1)/4

Case 2:
n = 2m+1
m+1 odd and m even numbers.
Total ways: (m+1)C2 + mC2
= 1/2[m.(m+1) + m.(m-1)] = m/2[2m] = m^2 = (n-1)^2/4

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