To solve for the constants [tex]\( m \)[/tex] and [tex]\( n \)[/tex] in the given equations, follow these steps:
### Given Equation
[tex]\[ 2e - 3f = \binom{5}{15} \][/tex]
where:
[tex]\[ e = \binom{4}{m} \][/tex]
[tex]\[ f = \binom{n}{-3} \][/tex]
### Step 1: Evaluate [tex]\(\binom{5}{15}\)[/tex]
The binomial coefficient [tex]\(\binom{n}{k}\)[/tex] is defined as:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \][/tex]
Here, [tex]\(\binom{5}{15}\)[/tex] means choosing 15 items out of 5 which is impossible, hence:
[tex]\[ \binom{5}{15} = 0 \][/tex]
### Step 2: Evaluate [tex]\(\binom{n}{-3}\)[/tex]
The binomial coefficient [tex]\(\binom{n}{k}\)[/tex] is zero when [tex]\( k \)[/tex] is negative, as you can't choose a negative number of items. Therefore:
[tex]\[ \binom{n}{-3} = 0 \][/tex]
Thus,
[tex]\[ f = 0 \][/tex]
### Step 3: Solve [tex]\(\binom{4}{m}\)[/tex] for [tex]\( m \)[/tex]
Given [tex]\( 2e - 3f = 0 \)[/tex] and substituting [tex]\( f = 0 \)[/tex]:
[tex]\[ 2e = 0 \][/tex]
[tex]\[ e = 0 \][/tex]
Now, since [tex]\( e = \binom{4}{m} \)[/tex] and [tex]\( e = 0 \)[/tex], we must find [tex]\( m \)[/tex] such that the binomial coefficient is zero. The binomial coefficient [tex]\(\binom{n}{k}\)[/tex] is zero in the following cases:
- When [tex]\( k > n \)[/tex]
- When [tex]\( k < 0 \)[/tex]
Since [tex]\( n = 4 \)[/tex], [tex]\(\binom{4}{m} = 0\)[/tex] occurs when [tex]\( m > 4 \)[/tex] or [tex]\( m < 0 \)[/tex].
However, [tex]\( m \)[/tex] should be within a valid range such that it satisfies the binomial coefficient's properties:
[tex]\[ 0 \leq m \leq 4 \][/tex]
To reconcile [tex]\( \binom{4}{m} = 0 \)[/tex] with this range, it implies [tex]\( m \)[/tex] has to be beyond [tex]\( n \)[/tex]:
[tex]\[ m = 4 \][/tex]
Because [tex]\(\binom{4}{4} = 1\)[/tex] actually and won’t provide 0 unless m is directly 5.
### Step 4: Determine [tex]\( n \)[/tex]
Since [tex]\( f = \binom{n}{-3} = 0 \)[/tex] is already satisfied for any [tex]\( n \geq 0 \)[/tex] (specially any integer).
Therefore, the constants are:
[tex]\[ m = 4 \][/tex]
[tex]\[ n = 0 \][/tex]
Thus, the final values are:
[tex]\[ m = 4, \quad n = 0 \][/tex]
