JEE 2023 — Mathematics PYQ

To solve this, we use the Stars and Bars method. However, the method typically requires variables to be non-negative (i.e., $\ge 0$). We need to transform our given variables.

Step 1: Substitute the variables

Let:

• $x_1=x-1⟹x=x_1+1$

• $y_1=y-3⟹y=y_1+3$

• $z_1=z-4⟹z=z_1+4$

Now, since $x\ge 1,y\ge 3,z\ge 4$, our new variables are all non-negative:

$x_1,y_1,z_1\ge 0$

Step 2: Rewrite the original equation

Substitute these into $x+y+z=21$:

Step 3: Apply the Stars and Bars formula

The number of non-negative integral solutions for an equation of the form $n_1+n_2+...+n_r=n$ is given by:

Here, $n=13$ and $r=3$ (the number of variables $x_1,y_1,z_1$).

Step 4: Calculate the final value

Correct Option: (B) 105