Let's solve the given problem step-by-step.
1. Calculate [tex]\( p \)[/tex]:
We need to determine the value of [tex]\( p \)[/tex], where
[tex]\[
p = \binom{-7}{1}
\][/tex]
The binomial coefficient [tex]\(\binom{n}{k}\)[/tex] represents the number of ways to choose [tex]\( k \)[/tex] objects from [tex]\( n \)[/tex] objects without regard to the order of selection. For any non-negative integer [tex]\( n \)[/tex], [tex]\(\binom{n}{k}\)[/tex] is defined as:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
However, the binomial coefficient for negative integers and/or when [tex]\( k > n \)[/tex] is defined to be 0. In our case, since [tex]\( n = -7 \)[/tex] is negative, we have:
[tex]\[
\binom{-7}{1} = 0
\][/tex]
2. Calculate [tex]\( q \)[/tex]:
Next, we determine the value of [tex]\( q \)[/tex], where
[tex]\[
q = \binom{11}{15}
\][/tex]
Here, [tex]\( k = 15 \)[/tex] is greater than [tex]\( n = 11 \)[/tex]. By the properties of binomial coefficients, when [tex]\( k > n \)[/tex], the value of the binomial coefficient is 0:
[tex]\[
\binom{11}{15} = 0
\][/tex]
3. Calculate [tex]\( 4p + q \)[/tex]:
With [tex]\( p = 0 \)[/tex] and [tex]\( q = 0 \)[/tex], we now calculate [tex]\( 4p + q \)[/tex]:
[tex]\[
4p + q = 4 \times 0 + 0 = 0
\][/tex]
4. Express the result as a column vector:
Finally, expressing the result in the form of a column vector, we get:
[tex]\[
\begin{pmatrix}
0
\end{pmatrix}
\][/tex]
So, the column vector representing [tex]\( 4p + q \)[/tex] is:
[tex]\[
\begin{pmatrix}
0
\end{pmatrix}
\][/tex]
