KCET · Maths · Binomial Theorem
The number \(\left(49^{2}-4\right)\left(49^{3}-49\right)\) is divisible by
- A 7 !
- B 9 !
- C \(6 !\)
- D 5 !
Answer & Solution
Correct Answer
(D) 5 !
Step-by-step Solution
Detailed explanation
\[
\begin{aligned}
&\text { Given, }\left(49^{2}-4\right)\left(49^{3}-49\right) \\
&\quad=\left[(49)^{2}-(2)^{2}\right]\left[(49)^{2}-1\right] \cdot 49 \\
&=(49+2)(49-2)(49+1)(49-1) \cdot 49 \\
&=51 \cdot 47 \cdot 50 \cdot 48 \cdot 49 \\
&=(51 \cdot 50 \cdot 49 \cdot 48 \cdot 47)
\end{aligned}
\]
Which is the product of five consecutive integers and hence it divisible by 5 !.
\begin{aligned}
&\text { Given, }\left(49^{2}-4\right)\left(49^{3}-49\right) \\
&\quad=\left[(49)^{2}-(2)^{2}\right]\left[(49)^{2}-1\right] \cdot 49 \\
&=(49+2)(49-2)(49+1)(49-1) \cdot 49 \\
&=51 \cdot 47 \cdot 50 \cdot 48 \cdot 49 \\
&=(51 \cdot 50 \cdot 49 \cdot 48 \cdot 47)
\end{aligned}
\]
Which is the product of five consecutive integers and hence it divisible by 5 !.
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