Question

From each box you can move only to the immediate right box or the immediate top box. You cannot move into or through a shaded box. How many ways are there to move from the box marked $S$ to the box marked $D$?

• A. $8$
• B. $10$
• C. $12$
• D. $14$

Answer 1

Answer: (C)

$12$

Let us first assume there are no shaded boxes then all that we need to do to get to D from S is

Take 3 steps up and 3 steps to the right in any order

No. of ways is nothing but choosing which 3 steps would be up(or right).

We can do this in ${}^6{C}_3=20$ ways.

Now if there are shaded boxes we can simply calculate no. of ways of reaching D via the shaded boxes.

Starting from S we need to take 3 steps up and 1 step to the right to reach the upper blockage. From there we can reach D with 2 right steps only.

Hence no. of ways = ${}^4{C}_3$ $\times$ ${}^2{C}_2=4$

Again starting from S we can reach right blockage in 2 right steps only and reach D from there with 3 steps up and 1 to the right.

So no. of ways = ${}^2{C}_2\times$ ${}^4{C}_3=4$

So total no. of ways = $20-(4+4)=12$