First slide

Combinations

Question

The number of ways in which three numbers in A.P. can be selected from 1, 2, 3, …, n is

Moderate

A

$\frac{n (n - 2)}{4}$ ,when n is even
B

$\frac{1}{4} (n - 1)^{2}$ when n is odd
C

$\frac{n (n - 2)}{2}$ when n is even
D

None of these

Solution

Given numbers are 1, 2, 3, … n.
Let the three selected numbers in A.P. be a, b, c, then
$b = \frac{a + c}{2} or a + c = 2 b - - - - (i)$
From (1) it is clear that a + c should be an even integer. This
is possible only when both a and c are odd or both are even.
Case I. When n is even. Let n = 2m
The number of odd numbers = m
and number of even numbers = m
∴ number of selections of a and c from m odd numbers = $^{m} C_{2}$
Number of selections of a and c from m even numbers = $^{m} C_{2}$
∴ Number of ways in this case = 2 ⋅ $^{m} C_{2}$ = m (m – 1)
$= \frac{n}{2} (\frac{n}{2} - 1) = \frac{n (n - 2)}{4}$
Case II. When n is odd. Let n = 2m + 1
Then, number of odd numbers = m + 1
and number of even numbers = m
∴ Required number in this case $=^{m + 1} C_{2} +^{m} C_{2}$
$\begin{aligned} = \frac{(m + 1) m}{2} + \frac{m (m - 1)}{2} = m^{2} = {(\frac{n - 1}{2})}^{2} \\ = \frac{1}{4} (n - 1)^{2} \end{aligned}$

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