
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
162.3k+ views
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.
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.
Recently Updated Pages
JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Chemical Properties of Hydrogen - Important Concepts for JEE Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NEET 2025 – Every New Update You Need to Know
