To solve the problem, we need to calculate the binomial coefficients [tex]\(\binom{2}{-3}\)[/tex] and [tex]\(\binom{0}{9}\)[/tex], then use their values to determine [tex]\(2a - 3b\)[/tex].
1. Understanding the Binomial Coefficient:
The binomial coefficient [tex]\(\binom{n}{k}\)[/tex] is defined as:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
However, there are conditions where the binomial coefficient is zero:
- If [tex]\(k < 0\)[/tex], [tex]\(\binom{n}{k} = 0\)[/tex].
- If [tex]\(k > n\)[/tex], [tex]\(\binom{n}{k} = 0\)[/tex].
2. Calculate [tex]\(a\)[/tex]:
Given [tex]\(a = \binom{2}{-3}\)[/tex]:
- Since [tex]\(-3 < 0\)[/tex], according to the condition mentioned, [tex]\(\binom{2}{-3} = 0\)[/tex].
Therefore, [tex]\(a = 0\)[/tex].
3. Calculate [tex]\(b\)[/tex]:
Given [tex]\(b = \binom{0}{9}\)[/tex]:
- Since [tex]\(9 > 0\)[/tex], according to the condition mentioned, [tex]\(\binom{0}{9} = 0\)[/tex].
Therefore, [tex]\(b = 0\)[/tex].
4. Calculate [tex]\(2a - 3b\)[/tex]:
- Substitute [tex]\(a = 0\)[/tex] and [tex]\(b = 0\)[/tex] into the expression [tex]\(2a - 3b\)[/tex]:
[tex]\[
2a - 3b = 2(0) - 3(0) = 0 - 0 = 0
\][/tex]
Hence, the value of [tex]\(2a - 3b\)[/tex] is [tex]\(0\)[/tex].
