Question

Find the number of integral solutions of the equation \[x + y + z = 20\] and \[x > - 1,y > - 2\] and \[z > - 3\].
A. \[{}^{25}{C_{23}}\]
B. \[{}^{17}{C_2}\]
C. \[{}^{23}{C_2}\]
D. None of these

Answer 1

Hint: In the given question, we need to find the number of integral solutions of the given equation according to boundaries of the variables. For this, we will convert these boundaries into inequalities. After that, we will assign them particular variables and also by simplifying and using the following formula, we will get the desired result.

Formula used: The following formula used for solving the given question.
The total possibilities is given by, \[{}^{n + r - 1}{C_{r - 1}}\]
Here, \[n\] is the total number of things and \[r\] is the number of things that need to be selected from total things.

Complete step by step solution: We know that the given equation is \[x + y + z = 20\]
Also, \[x > - 1,y > - 2\] and \[z > - 3\]
Here, \[x\] is a non-negative integer.
Let \[y + 2 = b\] and \[z + 3 = c\]
Now, we will find the values of \[x,y,\] and \[z\] from the above equations.
So, we get \[y = b - 1\], \[z = c - 2\] and also \[x = a\]
But we know that \[x + y + z = 20\]
Thus, we get \[a + b - 1 + c - 2 = 20\]
By simplifying, we get
\[a + b + c = 23\]
By applying the formula of total possibilities, we get
\[{}^{n + r - 1}{C_{r - 1}} = {}^{23 + 3 - 1}{C_{3 - 1}}\]
This gives,
\[{}^{n + r - 1}{C_{r - 1}} = {}^{25}{C_2}\]
Hence, we get \[{}^{n + r - 1}{C_{r - 1}} = {}^{25}{C_{23}}\]

Thus, Option (A) is correct.

Additional Information: Probability refers to possibility. A random event's occurrence is the subject of this area of mathematics. The range of the value is zero to one. Mathematics has incorporated probability to forecast the likelihood of various events.

Note: Many students make mistakes in the calculation part as well as writing the total possibility rule. This is the only way through which we can solve the example in the simplest way. Also, it is essential to analyze the result of \[{}^{n + r - 1}{C_{r - 1}}\] to get the desired result.