Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Combinatorics and Integers.
The highest power of 18 contained in \({50 \choose 25}\) is
Integers
Combinatorics
Exponents
Answer: 1
B.Stat Objective Problem 93
Challenges and Thrills of Pre-College Mathematics by University Press
here \({50 \choose 25}\)=\(\frac{50!}{(25!)^{2}}\)=\(\frac{(50)(49)(....)(26)}{(25)(24)(...)(1)}\)
\(=(2)^{13}(49)(47)(45)(43)(41)(39)(37)(35)(33)(31)(29)(27) \times \frac{1}{12!}\)
\(=(2)^{10}(49)(47)(15)(43)(41)(13)(37)(35)(11)(31)(29)\times \frac{1}{(12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)}[(2)^{3}(27)(9)(3)]\)
\(=(2)^{10}(49)(47)(15)(43)(41)(13)(37)(35)(11)(31)(29)\times \frac{1}{(12)(11)(10)(8)(7)(5)(4)(1)}[(2)(9)]\)gives a factor of \((18)^{1}\) then highest power of 18 is 1.
I.S.I. & C.M.I. Entrance Program
Taught by students and alumni of I.S.I. & C.M.I.