'r' rows and 'r+1' columns of squares = r(r+1) squares.
4th row(for that matter any row) has (r+1) squares
7th column(in fact any column) has r squares
They have 1 square in common.
So, number of squares in 4th row or 7th column = (r+1)+r-1 = 2r
So, remaining squares = r(r+1) - 2r = r^2 - r

or

Given, r > 10

consider r=11 and r+1 = 12
total squares = r(r+1)=132
4th row and 7th column = 11+12 = 23

total remaining squares = 109 + 1(common squarea) = 110

(r^2)-r = 110
(11)^2-11 = 110

The answer choices have been transcribed incorrectly.
They should read as I have posted them below:

sukriti2hats wrote:A rectangular game board is composed of identical squares arranged in a rectangular array of r rows and r+1 columns. The r rows are numbered from 1 through r and the r+1 columns are numbered from 1 through r+1. If r>10 then which of the following represents number of squares that are neither in the 4th row nor in the 7th column?

A) r^2 - r
B) r^2 + 1
C) r^2
D) r^2 - 1
E) r^2 + r

Ignore the constraint that r>10.
Let r=6, with the result that the number of rows = 6 and that the number of columns = r+1 = 6+1 = 7.
Let X = a square on the board.
The board looks as follows:

XXXXXXX
XXXXXXX
XXXXXXX
XXXXXXX
XXXXXXX
XXXXXXX

Total number of squares = 6*7 = 42.
The 4th row and the 7th column are composed of the squares in red.
Total number of squares in red = 12.
Thus:
Toral number of squares in NEITHER the 4th row nor the 7th column = 42-12 = 30. This is our target.

Now plug r=6 into the answers to see which yields our target of 30.
Only A works:
r² - r = 6² - 6 = 30.

The correct answer is A.