Question

Number of non-negative integral solutions of the equation \[ a+b+2c=22 \] is

• A. \(144\)
• B. \(121\)
• C. \(168\)
• D. \(99\)

Hint:

Number of non-negative solutions of \(a+b=n\) is \(n+1\).

Answer 1

The Correct Option is A

To find the number of non-negative integral solutions for the equation \( a + b + 2c = 22 \), we will use the concept of stars and bars in combination with managing constraints due to the coefficient of \( c \).

The equation given is:

a + b + 2c = 22

where \( a, b, \) and \( c \) are non-negative integers.

Let's redefine the equation by considering different values for \( c \) because it has a coefficient of 2.

Rewriting \( a + b = 22 - 2c \), we will find the possible values of \( c \) and for each, the solutions to the modified equation.

1. Determine \( c \): Since \( a, b, \) and \( c \) are non-negative, the highest possible value for \( 2c \) is 22. Thus:
   • \( c \) ranges from 0 to 11. (Because \( 2c \leq 22 \) and \( c \) must be an integer)
2. Express the equation in terms of \( a \) and \( b \):
   • For each value of \( c \), the equation becomes \( a + b = 22 - 2c \).
3. Number of solutions for each case: The equation \( a + b = n \) (where \( n \) is a non-negative integer) has \( n + 1 \) solutions.
   • Thus, for each valid \( c \), there are \( (22 - 2c) + 1 \) solutions.
4. Calculate the total number of solutions: Sum up the solutions for each possible \( c \).

We perform these calculations:

• For \( c = 0 \), solutions = 23
• For \( c = 1 \), solutions = 21
• For \( c = 2 \), solutions = 19
• ... continue reducing by 2 each step ...
• For \( c = 11 \), solutions = 1

Thus, sum these values:

23 + 21 + 19 + …….. + 1 = 144

This is an arithmetic series where the first term \( a = 23 \), the last term is 1, and the number of terms is 12 (since \( c \) ranges from 0 to 11).

The sum of an arithmetic series is given by:

\[\text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term})\]

Substituting the values, we get:

\[\text{Sum} = \frac{12}{2} \times (23 + 1) = 6 \times 24 = 144\]

Therefore, the number of non-negative integral solutions is 144.