m parallel lines cut n parallel lines giving rise to 60 parallelograms. What is the value of (m + n) ?
9
Understanding Parallelograms Formed by Parallel Lines
The question asks us to find the value of $(m + n)$ given that $m$ parallel lines intersecting $n$ parallel lines create 60 parallelograms. This is a common problem involving combinations in geometry.
Formula for Counting Parallelograms
When a set of $m$ parallel lines intersects another set of $n$ parallel lines, the number of parallelograms formed is determined by choosing two lines from the first set and two lines from the second set to form the sides of a parallelogram.
- To choose 2 lines from $m$ parallel lines, the number of ways is given by the combination formula $\binom{m}{2}$.
- To choose 2 lines from $n$ parallel lines, the number of ways is given by the combination formula $\binom{n}{2}$.
The total number of parallelograms is the product of these two combinations:
Number of parallelograms = $\binom{m}{2} \times \binom{n}{2}$
Setting up the Equation
We are given that the total number of parallelograms is 60. So, we can write the equation:
$\binom{m}{2} \times \binom{n}{2} = 60$
Solving the Equation
Recall the formula for combinations: $\binom{k}{2} = \frac{k(k-1)}{2}$.
Substituting this into our equation:
$\frac{m(m-1)}{2} \times \frac{n(n-1)}{2} = 60$
Multiply both sides by 4:
$m(m-1)n(n-1) = 240$
We need to find integer values for $m$ (where $m \ge 2$) and $n$ (where $n \ge 2$) such that the product of $m(m-1)$ and $n(n-1)$ is 240.
Let's look at the values of $k(k-1)$ for small integer values of $k$ ($k \ge 2$):
Value of k |
Value of k(k-1) |
|---|---|
| 2 | 2(1) = 2 |
| 3 | 3(2) = 6 |
| 4 | 4(3) = 12 |
| 5 | 5(4) = 20 |
| 6 | 6(5) = 30 |
| 7 | 7(6) = 42 |
| 8 | 8(7) = 56 |
| 9 | 9(8) = 72 |
We need to find two numbers from the second column that multiply to 240. Let's check the factors of 240:
- 1 x 240 (240 is not in the list $k(k-1)$ for reasonable k)
- 2 x 120 (120 is not in the list)
- 3 x 80 (80 is not in the list)
- 4 x 60 (60 is not in the list)
- 5 x 48 (48 is not in the list)
- 6 x 40 (40 is not in the list)
- 8 x 30 (30 is in the list: 6x5. 8 is not $k(k-1)$)
- 10 x 24 (Neither 10 nor 24 are in the list)
- 12 x 20 (Both 12 and 20 are in the list!)
We found a pair: 12 and 20.
- If $m(m-1) = 12$, then $m = 4$ (since 4 x 3 = 12).
- If $n(n-1) = 20$, then $n = 5$ (since 5 x 4 = 20).
Alternatively, it could be $m=5$ and $n=4$, as the formula is symmetric with respect to $m$ and $n$.
Let's verify with $m=4$ and $n=5$:
Number of parallelograms = $\binom{4}{2} \times \binom{5}{2} = \frac{4 \times 3}{2} \times \frac{5 \times 4}{2} = 6 \times 10 = 60$. This matches the given information.
Finding the Value of (m + n)
In either case ($m=4, n=5$ or $m=5, n=4$), the value of $(m + n)$ is:
$m + n = 4 + 5 = 9$
Conclusion
The value of $(m + n)$ is 9.
Revision Table: Counting Parallelograms
| Concept | Explanation |
|---|---|
| Parallel Lines Intersection | When two sets of parallel lines intersect, they can form parallelograms. |
| Choosing Lines | A parallelogram is formed by selecting two lines from one set and two lines from the other set. |
| Combination Formula | The number of ways to choose $k$ items from a set of $n$ is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. For choosing 2 lines, it's $\binom{n}{2} = \frac{n(n-1)}{2}$. |
| Total Parallelograms Formula | Given $m$ and $n$ sets of parallel lines, total parallelograms = $\binom{m}{2} \times \binom{n}{2}$. |
Additional Information: Geometric Combinations
This problem is an example of using combinations to count geometric shapes. Here are some related concepts:
- Lines formed by points: The number of lines formed by $n$ non-collinear points is $\binom{n}{2}$. If some points are collinear, you subtract the combinations of collinear points and add 1 for the single line they form.
- Triangles formed by points: The number of triangles formed by $n$ non-collinear points is $\binom{n}{3}$.
- Rectangles formed by grid lines: Similar to parallelograms, the number of rectangles in a grid formed by $m$ horizontal and $n$ vertical lines is $\binom{m}{2} \times \binom{n}{2}$. (Rectangles are a special case of parallelograms).
Understanding combinations is crucial for solving problems involving counting arrangements or selections of objects, including geometric figures formed by intersecting lines or sets of points.
Important Questions from Permutation and Combination
5-digit numbers are formed using the digits 0, 1, 2, 4, 5 without repetition. What is the percentage of numbers which are greater than 50,000 ?
In a race, there are 4 members in a team. Each member has to cover 5 km one after another. If the total time taken is 30 minutes, then what would have been the average speed?
If Quantity A is the number of ways to assign a number from 1 to 5 without repetition to each of four people, and Quantity B is the number of ways to assign a number from 1 to 5 without repetition to each of 5 people, then which of the following statements is correct with respect to Quantities A and B?
Which of the following muscles regulates the exit of food from the stomach into the small intestine?
There are 9 cups placed on a table arranged in equal number of rows and columns out of which 6 cups contain coffee and 3 cups contain tea. In how many ways can they be arranged so that each row should contain at least one cup of coffee?
