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Answer :
To solve the problem, we need to find the value of the expression [tex]\(\binom{4}{5}^6\)[/tex] and determine which of the given options is equal to this expression.
### Understanding the Binomial Coefficient
The binomial coefficient [tex]\(\binom{n}{k}\)[/tex], pronounced as "n choose k," represents the number of ways to choose [tex]\(k\)[/tex] elements from a set of [tex]\(n\)[/tex] elements without considering the order.
Mathematically, it is given by:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
where [tex]\(n!\)[/tex] (n factorial) is the product of all positive integers up to [tex]\(n\)[/tex].
### Applying this to [tex]\(\binom{4}{5}\)[/tex]
In [tex]\(\binom{4}{5}\)[/tex], we are trying to choose 5 items from a set of 4, which is impossible because you can't choose more items than are available. This results in:
[tex]\[
\binom{4}{5} = 0
\][/tex]
### Raising the Binomial Coefficient to a Power
The next part of the expression is raising this result to the power of 6:
[tex]\[
\binom{4}{5}^6 = 0^6
\][/tex]
### Calculating [tex]\(0^6\)[/tex]
Any non-positive number raised to any positive power is still 0. This means:
[tex]\[
0^6 = 0
\][/tex]
### Comparing to the Given Options
Now we compare this result (0) to the provided options:
A. [tex]\(6 \cdot\binom{4}{5}\)[/tex] \\
B. [tex]\(\frac{4^6}{5^8}\)[/tex] \\
C. [tex]\(\frac{24}{30}\)[/tex] \\
D. [tex]\(\frac{4^4}{5}\)[/tex]
Since we calculated that [tex]\(\binom{4}{5}^6 = 0\)[/tex], and none of the options equal 0, none of the options A through D matches the result of [tex]\(\binom{4}{5}^6\)[/tex], which equals 0.
### Understanding the Binomial Coefficient
The binomial coefficient [tex]\(\binom{n}{k}\)[/tex], pronounced as "n choose k," represents the number of ways to choose [tex]\(k\)[/tex] elements from a set of [tex]\(n\)[/tex] elements without considering the order.
Mathematically, it is given by:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
where [tex]\(n!\)[/tex] (n factorial) is the product of all positive integers up to [tex]\(n\)[/tex].
### Applying this to [tex]\(\binom{4}{5}\)[/tex]
In [tex]\(\binom{4}{5}\)[/tex], we are trying to choose 5 items from a set of 4, which is impossible because you can't choose more items than are available. This results in:
[tex]\[
\binom{4}{5} = 0
\][/tex]
### Raising the Binomial Coefficient to a Power
The next part of the expression is raising this result to the power of 6:
[tex]\[
\binom{4}{5}^6 = 0^6
\][/tex]
### Calculating [tex]\(0^6\)[/tex]
Any non-positive number raised to any positive power is still 0. This means:
[tex]\[
0^6 = 0
\][/tex]
### Comparing to the Given Options
Now we compare this result (0) to the provided options:
A. [tex]\(6 \cdot\binom{4}{5}\)[/tex] \\
B. [tex]\(\frac{4^6}{5^8}\)[/tex] \\
C. [tex]\(\frac{24}{30}\)[/tex] \\
D. [tex]\(\frac{4^4}{5}\)[/tex]
Since we calculated that [tex]\(\binom{4}{5}^6 = 0\)[/tex], and none of the options equal 0, none of the options A through D matches the result of [tex]\(\binom{4}{5}^6\)[/tex], which equals 0.
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