We know that the **no. of +ve integral solution to the equation $x_{1} + x_{2} + x_{3}+ ... + x_{k} = n$ is given by $\binom{n-1}{k-1}$**.

**(a)** Let, $x_{1}$ and $x_{2}$ be the two positive integers (not necessarily distinct) such that their sum is equal to $n\ (n≥2)$.

Therefore, $x_{1} + x_{2} = n$.

Now, we have to find all such values of $x_{1}$ and $x_{2}$ that satisfy the above equation. In other words, the problem basically reduces to finding the no. of +ve integral solution to the above equation.

Therefore, the req. no. of ways is $\binom{n-1}{2-1} = \binom{n-1}{1}$.

**(b)** Let, $x_{1}$, $x_{2}$ and $x_{3}$ be the three positive integers (not necessarily distinct) such that their sum is equal to $n\ (n≥3)$.

Therefore, $x_{1} + x_{2} + x_{3} = n$.

Now, we have to find all such values of $x_{1}$, $x_{2}$ and $x_{3}$ that satisfy the above equation. In other words, the problem basically reduces to finding the no. of +ve integral solution to the above equation.

Therefore, the req. no. of ways is $\binom{n-1}{3-1} = \binom{n-1}{2}$.

**(c)** Let, $x_{1}$, $x_{2}$, $x_{3}$,...,$x_{k}$ be the $k$ positive integers (not necessarily distinct) such that their sum is equal to $n\ (n≥k)$.

Therefore, $x_{1} + x_{2} + x_{3}+ ... + x_{k} = n$.

Now, we have to find all such values of $x_{1}$, $x_{2}$, $x_{3}$,...,$x_{k}$ that satisfy the above equation. In other words, the problem basically reduces to finding the no. of +ve integral solution to the above equation.

Therefore, the req. no. of ways is $\binom{n-1}{k-1}$.