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Given:
[tex]\[ p = \binom{-7}{1} \text{ and } q = \binom{11}{15} \][/tex]

Work out [tex]\( 4p + q \)[/tex].

Give your answer as a column vector.


Sagot :

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]