Question:
The number $(49^2 -4) (49^2 -49)$ is divisible by ..................
The number $(49^2 -4) (49^2 -49)$ is divisible by ..................
Updated On: May 17, 2024
- 6!
- 5!
- 7!
- 9!
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The Correct Option is B
Solution and Explanation
Given, $\left(49^{2}-4\right)\left(49^{3}-49\right)$
$=\left[(49)^{2}-(2)^{2}\right]\left[(49)^{2}-1\right] \cdot 49$
$=(49+2)(49-2)(49+1)(49-1) \cdot 49$
$=51 \cdot 47 \cdot 50 \cdot 48 \cdot 49$
$=(51 \cdot 50 \cdot 49 \cdot 48 \cdot 47)$
Which is the product of five consecutive integers and hence it divisible by $5 !$.
$=\left[(49)^{2}-(2)^{2}\right]\left[(49)^{2}-1\right] \cdot 49$
$=(49+2)(49-2)(49+1)(49-1) \cdot 49$
$=51 \cdot 47 \cdot 50 \cdot 48 \cdot 49$
$=(51 \cdot 50 \cdot 49 \cdot 48 \cdot 47)$
Which is the product of five consecutive integers and hence it divisible by $5 !$.
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Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .