In questions like these, observing patterns by calculating the first few terms of the sequence is important. IN fact, obtaining the pattern is the cornerstone of solving any problem on sequences.
When you see a question with a number of terms and the ellipsis symbols (the dots basically), know that the question is one on sequences. And at that instant, you should look at observing and developing a pattern.
Let’s look at evaluating the given expression.
P = 1! + (2x2!) + (3x3!) +…. + (11x11!). The fact that there is a 11x11! in the expression should tell us that it’s pointless to evaluate each expression. And that’s one of the hallmarks of a question on sequences, there will be one or two terms that look intimidating individually, but fit in nicely into the pattern if you can find one.
1! = 1
2X2! = 2x2 = 4
3X3! = 3x6 = 18.
No clue at this stage? Observe the expression. A fairly huge expression with lots of big numbers to be added. What if we bring in some subtraction and cancel out some terms? It will reduce the number of additions we have to do.
1! = 2! – 1!
2X2! = 4 = 3! – 2!
3X3! = 18 = 4! – 3!.
This is a neat little pattern developing here. So now you understand that a lot of terms will cancel out.
So, p = 2! – 1! + 3!-2! + 4! – 3! + 5! – 4!+……. + 12! – 11!. When we cancel out terms, we see that we will be left with p = 12! – 1!.
Therefore, p+2 = 12! + 1. When 12! + 1 is divided by 12!, clearly, the remainder is 1.
The correct answer option is B.
Hope that helps!