[Math] How many ways are there to select an ordered pair of numbers from $1$ to $7$ so that the sum is even

Question

How many ways are there to select an ordered pair of numbers from $1$ to $7$ so that the sum is even?

Soln: The way I tried approaching this problem was, I made a grid of $7 \times 7$ squares. So $49$ ordered pairs in total. Also the sum of two numbers is even only if the numbers selected are either both even or both odd. So, how do I proceed further?
Thank you.

Answer 1

If we are allowed to select each element once, then add up the following:

• The number of ways to select an ordered pair from $\{2,4,6\}$, which is $3\cdot2=6$
• The number of ways to select an ordered pair from $\{1,3,5,7\}$, which is $4\cdot3=12$

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If we are allowed to select each element twice, then add up the following:

• The number of ways to select an ordered pair from $\{2,4,6\}$, which is $3\cdot3=9$
• The number of ways to select an ordered pair from $\{1,3,5,7\}$, which is $4\cdot4=16$