Perfectly chosen

Taking only the units digit gives us (1x2x3x4 = 24) Hence taking units digit which is 4 and Adding 1 gives us (4+1=5)... Since units digit is 5 hence the logical answer should be 505.

Let's assume: $a=21$

Then,

$\sqrt{21\times22\times23\times24+1} = \sqrt{a(a+1)(a+2)(a+3)+1}$

If we solve the equation, we get:

$\sqrt{a(a+1)(a+2)(a+3)+1}=a^{2}+3a+1$

So, $a^{2}+3a+1=21^{2}+3\times21+1$

$= 441+63+1=505$

Thus, the answer is: $\sqrt{21\times22\times23\times24+1} = \boxed{505}$

This problem is simply based on the understanding of the order of operations (logical operators like addition, subtraction,etc..). There are two types of ways in which different countries prioritize the order of operations (PEMDAS and BODMAS/BEDMAS). In both PEMDAS and BODMAS, the multiplication part is done first and then addition gets preference. So, we can solve the problem like this ---->

$\sqrt{21\times 22\times 23\times 24 + 1}=\sqrt{462\times 552 + 1}= \sqrt{255024+1}= \sqrt{255025}=\boxed{\pm 505}$

But since here there are only positive number choices, so we take the positive value as the answer.

So, the answer is $=\boxed{505}$

To know more about order of operations, refer to this link : Order of Operations

There are infinitive ways to reach the destination(solution) of mathematics... keep going friends

Use scratchpad to answer!!! :p

WE can write under root,21X(21+1)(21+2)(21+4) +1,By solvingand taking squareroot,we get 21X21 +21+3 +1 =441+64 =505 Ans.

K>k>GARG,India

units digit will be 1 2 3*4=24 ,adding 1 then it will be 25 which ends with unit digit 5 hence considering options there is only one multiple of 5 which is the answer.....

just multiply 1x2x3x4 viz =24 n then add 1 = 24+1=25 therefore the digit at the units place has to be 5. Thus ans. is 505

This is when you take out the calculator... getting $\boxed{505}$