Question

The number of rectangles that can be found on chessboard is:
(1) $8{C_2} \cdot 8{C_2}$
(2) $64{C_4}$
(3) $9{C_2} \cdot 9{C_2}$
(4) $8{C_2}$

Answer 1

The given question requires application of combination technique. A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. It is denoted by the symbol $C$.

Complete step by step solution:
Combinations refer to the combination of $n$ things taken $k$ at a time without repetition. It can be solved using the following formula:
$C(n,k) = \dfrac{{n!}}{{(n - k)!k!}}$
Example: Picking a team of $3$ people from a group of $10$ can be solved as

$$C(10,3) = \dfrac{{10!}}{{(10 - 3)!3!}} = \dfrac{{10!}}{{7!3!}} \\\ C(10,3) = \dfrac{{10 \times 9 \times 8 \times 7!}}{{7!(3 \times 2 \times 1)}} = 120 \\\$$

Now to solve the given question, first we have to determine how the rectangle is formed, which we can easily conclude as it is made up of 2 parallel lines in the chessboard.
There will be a total of 8 $(n)$ boxes on the chessboard. Hence the total number of lines in the rectangle will be $n + 1$ which will give us $9$.
So, we can conclude that the total number of rectangles in the chessboard will be $9{C_2} \cdot 9{C_2}$.
We can use the shortcut formula $^{n + 1}{C_2}{ \cdot ^{n + 1}}{C_2}$ to find out the number of rectangles in $n \cdot n$ chessboard.
Hence option (3) is correct.

Note:

1. The solution is solved on the assumption that squares are included in “rectangle”.
2. Remember to split up the factorials and simplify to solve combination sums.
3. Combinations are used only when order is not important. If order is important, permutation will be used.
4. One should be familiar with common terms like number of king cards in the deck of cards, number of planets, number of pieces in chess etc. to solve sums relating to combinations.