Try this beautiful Problem based on Counting Principle from AMC 8, 2020 Problem 21.
A game board consists of 64 squares that alternate in color between black and white. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P$. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7 -step paths are there from $P$ to $Q$ ? (The figure shows a sample path.)
![[asy]//diagram by SirCalcsALot size(200); int[] x = {6, 5, 4, 5, 6, 5, 6}; int[] y = {1, 2, 3, 4, 5, 6, 7}; int N = 7; for (int i = 0; i < 8; ++i) { for (int j = 0; j < 8; ++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)); if ((i+j) % 2 == 0) { filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,black); } } } for (int i = 0; i < N; ++i) { draw(circle((x[i],y[i])+(0.5,0.5),0.35)); } label("$P$", (5.5, 0.5)); label("$Q$", (6.5, 7.5)); [/asy]](https://latex.artofproblemsolving.com/2/1/0/210b98fc88e4c297773b0400a31787193c0d4338.png)
Game problem
Chess board
combination
AMC 2020 PROBLEM 21
28
See that
the number of ways to move from P to that square is the sum of the numbers of ways to move from P to each of the white squares immediately beneath it
Try to construct a diagram and write the number of ways we can step onto that square from P which is calculated as the sum of the numbers on the white squares immediately beneath that square (and thus will represent the number of ways to remove from P to that square, as already stated).
So the diagram will look like,
![[asy] int N = 7; for (int i = 0; i < 8; ++i) { for (int j = 0; j < 8; ++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)); if ((i+j) % 2 == 0) { filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,black); } } } label("$1$", (5.5, .5)); label("$1$", (4.5, 1.5)); label("$1$", (6.5, 1.5)); label("$1$", (3.5, 2.5)); label("$1$", (7.5, 2.5)); label("$2$", (5.5, 2.5)); label("$1$", (2.5, 3.5)); label("$3$", (6.5, 3.5)); label("$3$", (4.5, 3.5)); label("$4$", (3.5, 4.5)); label("$3$", (7.5, 4.5)); label("$6$", (5.5, 4.5)); label("$10$", (4.5, 5.5)); label("$9$", (6.5, 5.5)); label("$19$", (5.5, 6.5)); label("$9$", (7.5, 6.5)); label("$28$", (6.5, 7.5)); [/asy]](https://latex.artofproblemsolving.com/3/8/d/38df398e31d5f4a69999da41b16ceb51aef50425.png)
Try this beautiful Problem based on Counting Principle from AMC 8, 2020 Problem 21.
A game board consists of 64 squares that alternate in color between black and white. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P$. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7 -step paths are there from $P$ to $Q$ ? (The figure shows a sample path.)
Game problem
Chess board
combination
AMC 2020 PROBLEM 21
28
See that
the number of ways to move from P to that square is the sum of the numbers of ways to move from P to each of the white squares immediately beneath it
Try to construct a diagram and write the number of ways we can step onto that square from P which is calculated as the sum of the numbers on the white squares immediately beneath that square (and thus will represent the number of ways to remove from P to that square, as already stated).
So the diagram will look like,
