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BINOMIAL THEOREM — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSBINOMIAL THEOREM1 Mark
MCQ
If coefficient of $x^{100}$ in $1+(1+x)(1+x)^2+\ldots . .+(1+x)^n$ (if $n \geq 100$ ) is $C_{101}^{201}$ then the value of $n$ equals.
  • A
    202
  • B
    100
  • ✓
    200
  • D
    201
✓

Answer

Correct option: C.
200
  1. 200
Solution:
${ }^n C_r+{ }^n C_{(r+1)}={ }^{(n+1)} C_{(r+1)}$
coefficient of $x^{100}$ is ${ }^{100} \mathrm{C}_{100}+{ }^{101} \mathrm{C}_{100}+{ }^{102} \mathrm{C}_{100}+\ldots . . . .+{ }^n \mathrm{C}_{100}$.
Which is equal to ${ }^{(n+1)} \mathrm{C}_{101}$.
Therefore, $\mathrm{n}+1=201$
Which implies $n=200$

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