Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Combinatorics and Integers.

Combinatorics and Integers (B.Stat Objective Question)


The highest power of 18 contained in \({50 \choose 25}\) is

  • 104
  • 1
  • 1154
  • none of these

Key Concepts


Integers

Combinatorics

Exponents

Check the Answer


But try the problem first…

Answer: 1

Source
Suggested Reading

B.Stat Objective Problem 93

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

here \({50 \choose 25}\)=\(\frac{50!}{(25!)^{2}}\)=\(\frac{(50)(49)(….)(26)}{(25)(24)(…)(1)}\)

Second Hint

\(=(2)^{13}(49)(47)(45)(43)(41)(39)(37)(35)(33)(31)(29)(27) \times \frac{1}{12!}\)

Final Step

\(=(2)^{10}(49)(47)(15)(43)(41)(13)(37)(35)(11)(31)(29)\times \frac{1}{(12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)}[(2)^{3}(27)(9)(3)]\)

\(=(2)^{10}(49)(47)(15)(43)(41)(13)(37)(35)(11)(31)(29)\times \frac{1}{(12)(11)(10)(8)(7)(5)(4)(1)}[(2)(9)]\)gives a factor of \((18)^{1}\) then highest power of 18 is 1.

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