Statement-1: If n is the odd, integer number of ways in which three numbers are AP can be selected from 1, 2, 3, ………. n is (n−1)24Statement-2: AM of two odd numbers or two even number is an integer.

By selecting two odd numbers or two even numbers let (a,c) , we will have an AP a, a+c2, c 1,3,5,........,n→n+12 odd no ⇒ n+1 2C2 ways to select two numbers2,4,6,.......,n−1→n−12 even no's⇒ n−1 2C2 ways to select two numbersTotal no.of AP’s n+1 2C2+n−1 2C2

Statement-1: If n is the odd, integer number of ways in which three numbers are AP can be selected from 1, 2, 3, ………. n is (n−1)24Statement-2: AM of two odd numbers or two even number is an integer.

By selecting two odd numbers or two even numbers let (a,c) , we will have an AP a, a+c2, c 1,3,5,........,n→n+12 odd no ⇒ n+1 2C2 ways to select two numbers2,4,6,.......,n−1→n−12 even no's⇒ n−1 2C2 ways to select two numbersTotal no.of AP’s n+1 2C2+n−1 2C2

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Statement-1: If n is the odd, integer number of ways in which three numbers are AP can be selected from 1, 2, 3, ………. n is $\frac{{(n - 1)}^{2}}{4}$
Statement-2: AM of two odd numbers or two even number is an integer.

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Statement –1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1

Statement –1 is True, Statement – 2 is True; Statement – 2 is NOT a correct
explanation for Statement – 1

Statement –1 is True, Statement – 2 is False

Statement –1 is False, Statement – 2 is True

answer is A.

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Detailed Solution

By selecting two odd numbers or two even numbers let $(a, c)$ , we will have an AP $a, \frac{a + c}{2}, c$
$1, 3, 5, ........, n \to \frac{n + 1}{2} o d d n o \Rightarrow \frac{n + 1}{^{2} C_{2}}$ ways to select two numbers
$2, 4, 6, ......., n - 1 \to \frac{n - 1}{2} e v e n n o' s \Rightarrow \frac{n - 1}{^{2} C_{2}}$ ways to select two numbers
Total no.of AP’s $\frac{n + 1}{^{2} C_{2}} + \frac{n - 1}{^{2} C_{2}}$