To solve the system of equations given, we will proceed step-by-step to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
The given equations are:
1. [tex]\( 5 = \binom{4x - 4}{1 - y} \)[/tex]
2. [tex]\( t = \binom{20 + 2x}{17 - 3y} \)[/tex]
3. [tex]\( s = t \)[/tex] (which just means [tex]\( s \)[/tex] and [tex]\( t \)[/tex] are equal)
### Step 1: Solve the Binomial Coefficient Equation
We need to simplify and solve the first equation:
[tex]\[ 5 = \binom{4x - 4}{1 - y} \][/tex]
The binomial coefficient [tex]\( \binom{n}{k} \)[/tex] represents the number of ways to choose [tex]\( k \)[/tex] elements from [tex]\( n \)[/tex] elements and is given by:
[tex]\[ \binom{n}{k} = \frac{n!}{k! (n - k)!} \][/tex]
We need to find values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] such that this equality holds true. The possible values need to be integers since binomial coefficients are defined only for integer values.
Possible values of [tex]\( \binom{n}{k} \)[/tex] yielding 5 are reasonably small. We will check for low [tex]\( n \)[/tex] and [tex]\( k \)[/tex]. Typical pairs are:
- [tex]\( \binom{5}{1} = 5 \)[/tex]
- Any higher combinations should fit the required variables' constraints.
Thus, we can equate:
[tex]\[ 4x - 4 = 5 \quad \text{and} \quad 1 - y = 1 \][/tex]
From [tex]\( 4x - 4 = 5 \)[/tex]:
[tex]\[ 4x - 4 = 5 \][/tex]
[tex]\[ 4x = 9 \][/tex]
[tex]\[ x = \frac{9}{4} \][/tex]
From [tex]\( 1 - y = 1 \)[/tex]:
[tex]\[ y = 0 \][/tex]
Thus, we have determined that:
[tex]\[ x = \frac{9}{4} \quad \text{and} \quad y = 0 \][/tex]
### Step 2: Verification with the Second Equation
We substitute [tex]\( x = \frac{9}{4} \)[/tex] and [tex]\( y = 0 \)[/tex] into the second equation and see if it holds. Ensure these values make sense and are valid:
[tex]\[ t = \binom{20 + 2x}{17 - 3y} \quad \Rightarrow \quad t = \binom{20 + 2 \cdot \frac{9}{4}}{17 - 3 \cdot 0} \][/tex]
[tex]\[ t = \binom{20 + \frac{18}{4}}{17} = \binom{25.5}{17} \][/tex]
Since it requires integer values, there might be necessary integer-intuition verifications to see whether any miscalculation or consideration through approximate or non-viable continuous solutions holds because these steps show solutions essentially valid but doesn't convert optimally for qualitative expectations needing rectifications on assumptions possibly not aligning system adequately.
The value of [tex]\( x\)[/tex] and [tex]\( y\)[/tex] needs different alignment per:
```rechecking aligning workable boundaries under logical values intuitive pursuit rational considerations per requirement algebraic closeness applied \)
Conclusion: Re-review is needed `defining casing valid combinations integer integrity versatile fitting desired expectations rational solving [tex]\(an improved aligned procedural expected fits\)[/tex]` or interpretation needs flexible look insights correct definitions. Expected exhausted validated closely fitting \( revaluations towards close fitted final confirmable solution}
