Question

In each of its move, a pawn in a chess board can move one step either horizontally or vertically to its adjacent cell from its current position. If a pawn is initially located at the South-West corner cell of the chess board, then the number of ways it can reach the North-East corner cell with minimum number of moves, is

• A. ${}^{64}C_2$
• B. $2\times{}^{8}C_2$
• C. ${}^{14}C_7$
• D. ${}^{16}C_8$

Hint:

Shortest path problems on grids are solved using combinations: \[ {}^{m+n}C_m \] where $m$ and $n$ are moves in the two directions.

Answer 1

The Correct Option is C

Step 1: Picture the board.
A chessboard is an $8\times8$ grid of cells. Moving from the South-West corner to the North-East corner, the pawn must shift across to the far side and up to the top.
Step 2: Count the steps needed.
To cross $8$ columns the pawn takes $7$ steps right. To climb $8$ rows it takes $7$ steps up.
Step 3: Find the smallest total.
So the minimum number of moves is $7+7=14$, with no wasted moves.
Step 4: See the choice we make.
Every shortest path is just an order of $7$ right moves and $7$ up moves mixed together.
Step 5: Count the orders.
Out of the $14$ move slots, we only choose which $7$ are the right moves. That count is ${}^{14}C_7$.
Step 6: State the answer.
So the number of shortest paths is ${}^{14}C_7$. \[ \boxed{{}^{14}C_7} \]