Each of the sides of a quadrilateral is a natural number and the perimeter of the quadrilateral is always 12 units. Find the number of combinations of various lengths of the sides of the quadrilateral.

Options

A. 8
B. 10
C. 5
D. none of these

Answer

To find the number of combinations of various lengths of the sides of a quadrilateral with a perimeter of 12 units, we can denote the sides of the quadrilateral as \( a, b, c, \) and \( d \). Since the perimeter is the sum of all sides, we have:

\[ a + b + c + d = 12 \]

where \( a, b, c, \) and \( d \) are natural numbers (i.e., positive integers).

### Step 1: Setting Up the Equation
We need to find the number of solutions to the equation \( a + b + c + d = 12 \) under the condition that \( a, b, c, d \) are all natural numbers.

### Step 2: Transforming the Variables
To simplify the problem, we can transform the variables to account for the fact that they must be natural numbers. We can do this by letting:

\[ a' = a - 1, \quad b' = b - 1, \quad c' = c - 1, \quad d' = d - 1 \]

where \( a', b', c', d' \) are non-negative integers (i.e., they can be zero). This transformation gives us:

\[ (a' + 1) + (b' + 1) + (c' + 1) + (d' + 1) = 12 \]

which simplifies to:

\[ a' + b' + c' + d' = 8 \]

### Step 3: Applying the Stars and Bars Theorem
Now, we need to find the number of non-negative integer solutions to the equation \( a' + b' + c' + d' = 8 \). We can use the "stars and bars" theorem, which states that the number of ways to distribute \( n \) indistinguishable objects (stars) into \( k \) distinguishable boxes (the variables) is given by:

\[ \binom{n + k - 1}{k - 1} \]

In our case, \( n = 8 \) (the total we want to achieve) and \( k = 4 \) (the number of sides). Thus, we need to calculate:

\[ \binom{8 + 4 - 1}{4 - 1} = \binom{11}{3} \]

### Step 4: Calculating the Binomial Coefficient
Now, we calculate \( \binom{11}{3} \):

\[ \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = \frac{990}{6} = 165 \]

### Conclusion
Therefore, the number of combinations of various lengths of the sides of the quadrilateral is **165**.

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