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Binomial Theorem Part 3

Section: Mathematics

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Question : 43 of 53

Marks: +1, -0

The coefficient of

x^{70}

x^2(1+x)^{98}+x^3(1+x)^{97}+

x^4(1+x)^{96}+\dots+x^{54}(1+x)^{46}

{}^{99}C_{p}-{}^{46}C_{q}

Then a possible value to

p+q

[9 Apr 2024 Shift 1]

55
61
68
83

Solution: 👈: Video Solution

\;x^2(1+x)^{98}+x^3(1+x^{97})+x^4(1+x)^{96}+\dots

\;x^{54}(1+x)^{46}

Coeff. of

x^{70}:{}^{98}C_{68}+{}^{97}C_{67}+{}^{96}C_{66}+\dots

.

\;{}^{47}C_{17}+{}^{46}C_{16}

\;={}^{46}C_{30}+{}^{47}C_{30}+\dots+{}^{98}C_{30}

\;=({}^{46}C_{31}+{}^{46}C_{30})+{}^{47}C_{30}+\dots+{}^{98}C_{30}-{}^{46}C_{31}

\;={}^{47}C_{31}+{}^{47}C_{30}+\dots+{}^{98}C_{30}-{}^{46}C_{31}

\;\dots

\;={}^{99}C_{31}-{}^{46}C_{31}={}^{99}C_{p}-{}^{46}C_{q}

Possible values of

(p+q)

are

62,83,99,46

\Rightarrow p+q=83

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