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Algebra Arithmetic Combinatorics I.S.I. and C.M.I. Entrance ISI Entrance Videos

Binomial Expression | TOMATO B.Stat Objective 117

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Binomial Expression. You may use sequential hints to solve the problem.

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Binomial Expression.

Binomial Expression ( B.Stat Objective Question )


The coefficient of \(x^{2}\) in the binomial expression of \((1+x+x^{2})^{10}\) is

  • \({10 \choose 1}\)
  • \({10 \choose 1}+{10 \choose 2}\)
  • 0
  • none of these

Key Concepts


Cefficient

Binomials

Combinations

Check the Answer


But try the problem first…

Answer:\({10 \choose 1}+{10 \choose 2}\).

Source
Suggested Reading

B.Stat Objective Problem 117

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

\((\frac{1-x^{3}}{1-x})^{10}\)

=\((1-x^{3})^{10}(1-x)^{-10}\)

=\((1-{10 \choose 1}x^{3}\)

\(+{10 \choose 2}x^{6}+….)(1-10(-x)\)

\(+\frac{(-10)(-10)-1}{2}(-x)^{2}+…..)\)

Second Hint

coefficient of \(x^{2}\)=\(\frac{(-10)((-10)-1)}{2}\)

Final Step

=\({10 \choose 1}+{10 \choose 2}\).

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