Animated Solution for Mathematics - Permutations and Combinations: Number of integral solutions to the equation $x+y+z=21$, where $x\ge 1,y\ge 3,z\ge 4$, is equal to ____.

Imagine you are standing in front of a magical candy shop. You have $21$ identical candies, and you need to distribute them among three friends: $x$, $y$, and $z$.

There is a social contract: friend $x$ must receive at least $1$ candy, friend $y$ must receive at least $3$, and friend $z$ must receive at least $4$. We seek the number of ways to distribute these candies under these constraints.

When we look at the equation $x+y+z=21$, our instinct might be to jump straight to the 'Stars and Bars' formula. However, the standard formula $\left(\genfrac{}{}{0ex}{}{n+r-1}{r-1}\right)$ is a perfectionist.

It only works when your variables are non-negative, meaning $x,y,z\ge 0$. Because our variables are restricted ($x\ge 1$, $y\ge 3$, and $z\ge 4$), we must liberate our variables from these constraints to avoid counting physically impossible scenarios.

To solve this, we use a technique called 'pre-allocation.' We define new variables, $x^{\prime }$, $y^{\prime }$, and $z^{\prime }$, which represent the 'extra' candies each friend receives beyond their minimum requirement.

We set the following substitutions: $x=x^{\prime }+1$ $y=y^{\prime }+3$ $z=z^{\prime }+4$

Since $x\ge 1$, it follows that $x^{\prime }+1\ge 1$, which simplifies to $x^{\prime }\ge 0$. The same logic applies to $y^{\prime }$ and $z^{\prime }$, successfully transforming our constrained variables into non-negative ones.

Now, let us substitute these into our original equation:

If we group the constants, we get:

Subtracting $8$ from both sides, we arrive at the simplified equation:

This is the heart of the problem. We are now distributing $13$ candies among $3$ friends with no restrictions other than that they must receive a non-negative amount.

We are ready for the 'Stars and Bars' theorem. Here, $n=13$ (the remaining candies) and $r=3$ (the number of friends).

The formula is $\left(\genfrac{}{}{0ex}{}{n+r-1}{r-1}\right)$, which becomes:

Calculating this is straightforward:

There you have it! By respecting the constraints and transforming the problem, we found that there are exactly $105$ ways to satisfy the conditions. Remember, in the JEE, the most complex problems are often just simple problems wearing a disguise.