Additional Concepts
-$$1 - \frac { 5 ^ { P _ { 2 } } } { { } _ { 8 } P _ { 5 } }$$
A) $\frac { 335 } { 336 }$
B) $\frac { 1 } { 336 }$
C) $\frac { 23 } { 28 }$
D) $\frac { 337 } { 336 }$

Question

Quizplus · Accepted Answer

We first evaluate $P_2$ and $P_5$: $$P_2 = \frac{8!}{(8-2)!} = 56$$ $$P_5 = \frac{8!}{(8-5)!} = 6720$$ Then we plug in the values to the given expression: $$1 - \frac{5^{56}}{6720} = 1 - \frac{5^{56}}{2^6\cdot3\cdot5\cdot7} = \frac{2^6\cdot3\cdot5\cdot7-5^{56}}{2^6\cdot3\cdot5\cdot7}$$ Notice that $2^6\cdot3\cdot5\cdot7$ has a factor of $5$, so we can rewrite the expression as: \begin{align*} \frac{2^6\cdot3\cdot5\cdot7-5^{56}}{2^6\cdot3\cdot5\cdot7} &= \frac{2^6\cdot3\cdot5\cdot7-5^{56}\cdot5^{-56}}{2^6\cdot3\cdot5\cdot7\cdot5^{-1}} \ &= \frac{1}{5}\left(\frac{2^6\cdot3\cdot7}{2^{56}}-\frac{1}{5^{56}}\right) \ &= \frac{1}{5}\left(\frac{21}{2^{50}}-\frac{1}{5^{56}}\right) \end{align*} Since $2^{50}=4^{25}<5^{25}<5^{56}$, we have: $$\frac{21}{2^{50}}-\frac{1}{5^{56}} > \frac{21}{5^{25}}-\frac{1}{5^{56}} > 0$$ Therefore, the expression is greater than $0$ and less than $\frac{1}{5}$. The only choice that satisfies this condition is $\boxed{\textbf{(A)}\ \frac{335}{336}}$.

Additional Concepts - 1−5P28P51 - \frac { 5 ^ { P _ { 2 } } } { { } _ { 8 } P _ { 5 } }1−8​P5​5P2​​

Additional Concepts
- $1 - \frac { 5 ^ { P _ { 2 } } } { { } _ { 8 } P _ { 5 } }$