Permutations & Combinations
Highest Power of Prime in Factorial
Grade None
Question:
<p>Number of cyphers (trailing zeros) at the end of \(\binom{2016}{1008}\).</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: Find the highest power of 5 dividing a binomial coefficient using Legendre's formula applied to factorials.
<p><strong>Solution:</strong></p><p>$\binom{2016}{1008} = \frac{2016!}{(1008!)^2}$</p><p>$E_5(2016!) = \lfloor 2016/5 \rfloor + \lfloor 2016/25 \rfloor + \lfloor 2016/125 \rfloor + \lfloor 2016/625 \rfloor = 403 + 80 + 16 + 3 = 502$</p><p>$E_5(1008!) = \lfloor 1008/5 \rfloor + \lfloor 1008/25 \rfloor + \lfloor 1008/125 \rfloor + \lfloor 1008/625 \rfloor = 201 + 40 + 8 + 1 = 250$</p><p>Number of trailing zeros = $502 - 250 - 250 = 2$</p>
Correct Answer: C