Questions: Find all the solutions to the equation 3x + 4y = 60 in positive integers.

Find all the solutions to the equation 3x + 4y = 60 in positive integers.
Transcript text: Find all the solutions to the equation $3 x+4 y=60$ in positive integers.
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Solution

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Solution Steps

To solve the given problems:

  1. Combination of 117 and 247: This likely refers to the number of ways to choose 117 items from 247, which is a combinatorial problem. We can use the binomial coefficient formula \( \binom{n}{k} \).

  2. Find all the solutions to the equation \(3x + 4y = 60\) in positive integers: We need to find pairs \((x, y)\) such that both \(x\) and \(y\) are positive integers and satisfy the equation.

Solution Approach
  1. Use the binomial coefficient formula to calculate the combination.
  2. Iterate through possible values of \(x\) and solve for \(y\) to find all pairs that satisfy the equation.
Step 1: Calculate the Combination of 117 and 247

To find the number of ways to choose 117 items from 247, we use the binomial coefficient formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \( n = 247 \) and \( k = 117 \). The result is: \[ \binom{247}{117} = 8156776644148081679783442913085917667048289409413000753018125407845132279 \]

Step 2: Find All Solutions to the Equation \(3x + 4y = 60\) in Positive Integers

We need to find pairs \((x, y)\) such that both \(x\) and \(y\) are positive integers and satisfy the equation \(3x + 4y = 60\).

By iterating through possible values of \(x\) and solving for \(y\), we find the following solutions: \[ \begin{align_} x = 4, & \quad y = 12 \\ x = 8, & \quad y = 9 \\ x = 12, & \quad y = 6 \\ x = 16, & \quad y = 3 \\ \end{align_} \]

Final Answer

\[ \boxed{ \begin{aligned} &\text{Combination of 117 and 247:} \\ &8156776644148081679783442913085917667048289409413000753018125407845132279 \\ &\text{Solutions to the equation } 3x + 4y = 60 \text{ in positive integers:} \\ &(4, 12), (8, 9), (12, 6), (16, 3) \end{aligned} } \]

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