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June 10, 2020

Combinatorics and Integers | TOMATO B.Stat Objective 93

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


Answer: 1

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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