MyQuestionIcon

3

You visited us 3 times! Enjoying our articles? Unlock Full Access!

Sum of Product of Binomial Coefficients

The value of ...

Question

The value of
$(\frac{30}{0}) (\frac{30}{10}) - (\frac{30}{1}) (\frac{30}{11}) + (\frac{30}{2}) (\frac{30}{12}) . . . + (\frac{30}{20}) (\frac{30}{30})$ is; where $(\frac{n}{r}) =^{n} C_{r}$

A

(_{10}^{30})

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

B

(_{15}^{30})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

C

(_{30}^{60})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

D

(_{10}^{31})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $(_{10}^{30})$
To find
$^{30} C_{0}^{30} C_{10} -^{30} C_{1}^{30} C_{11} +^{30} C_{2}^{30} C_{12} - . . . +^{30} C_{20}^{30} C_{30}$
We know that
$(1 + x)^{30} =^{30} C_{0} +^{30} C_{1} x +^{30} C_{2} x^{2} + . . . . . +^{30} C_{20} x^{20} + . . .^{30} C_{30} x^{30} . . . . . (1) (x - 1)^{30} =^{30} C_{0} x^{30} -^{30} C_{1} x^{29} + . . . . +^{30} C_{10} x^{20} -^{30} C_{11} x^{19} +^{30} C_{12} x^{18} + . . .^{30} C_{30} x^{0} . . . . . (2)$
Multiplying equation (1) and (2),we get
$(x^{2} - 1)^{30} = () \times ()$
Equating the coefficients of $x^{20}$ on both sides, we get
$^{30} C_{10} =^{30} C_{0}^{30} C_{10} -^{30} C_{1}^{30} C_{11} +^{30} C_{2}^{30} C_{12} - . . . +^{30} C_{20}^{30} C_{30}$
$∴$ Req.value is $^{30} C_{10}$

Suggest Corrections

thumbs-up

0

Similar questions

Q. The value of

(\frac{30}{0}) (\frac{30}{10}) + (\frac{30}{1}) (\frac{30}{11}) + (\frac{30}{2}) (\frac{30}{12}) + . . . + (\frac{30}{20}) (\frac{30}{30})

(\frac{n}{r}) = n_{C_{r}}

Q. The value of

(\begin{matrix} 300 \end{matrix}) (\begin{matrix} 3010 \end{matrix}) - (\begin{matrix} 301 \end{matrix}) (\begin{matrix} 3011 \end{matrix}) + (\begin{matrix} 302 \end{matrix}) (\begin{matrix} 3012 \end{matrix}) - . . . + (\begin{matrix} 3020 \end{matrix}) (\begin{matrix} 3030 \end{matrix})

(\begin{matrix} n r \end{matrix}) = {^{n} C}_{r}

Q. The value of

(\begin{matrix} 300 \end{matrix})

(\begin{matrix} 3010 \end{matrix})

(\begin{matrix} 301 \end{matrix})

(\begin{matrix} 3011 \end{matrix})

(\begin{matrix} 302 \end{matrix})

(\begin{matrix} 3012 \end{matrix})

(\begin{matrix} 3020 \end{matrix})

(\begin{matrix} 3030 \end{matrix})

(\begin{matrix} n r \end{matrix})

^{n} C_{r} .

Q. The value of

^{30} C_{0}^{30} C_{10} -^{30} C_{1}^{30} C_{11} +^{30} C_{2}^{30} C_{12} . . . . . +^{30} C_{10}^{30} C_{20}

The value of $^{30} C_{0}$ $^{30} C_{1} 0$ - $^{30} C_{1}$ $^{30} C_{11}$ + $^{30} C_{2}$ $^{30} C_{12}$ - .......... + $^{30} C_{20}$ $^{30} C_{30}$ is

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Sum of Product of Binomial Coefficients

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Sum of Product of Binomial Coefficients

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon