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 $r>10$, which of the following represents the number of squares on board that are neither in the 4th row nor in the 7th column?
a) $r^2 - r$
b) $r^2 - 1$
c) $r^2 + r$
My approachI approached in this way: First to select any one row I have ${r\choose 1}$ ways. Then I selected one of the $r+1$ columns in ${r+1 \choose 1 }$ ways. So multiplying them, I was getting $r(r+1)$.
$\endgroup$ 13 Answers
$\begingroup$If we were just counting all of the squares in the board game we could use multiplication: r rows times r+1 columns. This would give us $r^2+r$ squares. If we take one row out (doesn't matter if its the first, second, third, etc) then we would have r-1 rows times r+1 columns. If we also then take a column out, we would have r-1 rows times r columns. So the total number of squares without a column and without a row is $r(r-1)=r^2-r$ so letter a is the answer.
$\endgroup$ $\begingroup$There are $r(r+1)$ squares total.
There are $r+1$ squares in the fourth row.
There are $r$ squares in the seventh column.
There is $1$ square in the fourth row AND the seventh column.
Use inclusion-exclusion from here.
$\endgroup$ $\begingroup$We are required to find the number of squares which are neither in 4th row nor in 7th column.
So, answer = Total squares on board - total squares in 4th row and 7th column
Total squares on the board = r (r + 1) = r^2 + r ----- {1}
squares in 4th row = r+1
squares in 7th column = r
But one is repeated square - on the 4th row and 7th column. Thus, total squares in 4th row and 7th column = (r+1) + (r) - 1 = 2r
So, from equation {1} : answer = r^2 + r - 2r = r^2 - r
Hope this helps !
$\endgroup$ 1
