Question

Statement: 1 remainder when ${{99}^{100}}$ is divided by 10 is 2 because
Statement: 2$^{n}{{C}_{r}}{{=}^{n}}{{C}_{n-r}}$
A.Statement 1 is true, statement 2 is true
Statement 2 is correct explanation for statement 1
B.Statement 1 is true statement 2 is true, statement2 is not a correct explanation for statement 1
C.Statement 1 is true, statement 2 is false
D.Statement 1 is false, statement 1 is true

Answer 1

Hint: 1st is to check even & odd power of 9 at unit place & 2nd statement is the rule for combination that if $^{n}{{C}_{x}}{{=}^{x}}{{C}_{y}}$ then either x=y or $x+y=n$

Formula used:
1st Statement we have ${{9}^{1}}=9\And {{9}^{2}}=81$
9 at unit place when power is odd & 1 at unit place if power of 9 is even.
In 2nd Statement we have$^{n}{{C}_{r}}{{=}^{n}}{{C}_{y}}$ then x=y or $x+y=n$
$^{n}{{C}_{r}}=\frac{n!}{\left( n-r \right)!r!}\text{ 0}\le \text{r}\le \text{n}$

Complete step-by-step answer:
 ${{99}^{100}}$ we check the unit digit
For
$\begin{align}
  & {{9}^{1}}=9 \\
 & {{9}^{2}}=81 \\
\end{align}$
1 at unit place
9 at unit place
${{9}^{3}}=9$ at unit place \[{{9}^{4}}=1\] at unit place
So for away odd powers of 9 we have 1 unit. So first statement is false
Statement2: $^{n}{{C}_{r}}{{=}^{n}}{{C}_{n-r}}$
LHS
$^{n}{{C}_{r}}=\frac{n!}{\left( n-r \right)!n!}\text{ 0=}r\le n$ (general formula)
RHS
$\begin{align}
  & ^{n}{{C}_{r}}=\frac{n!}{\left[ n-\left( n-r \right) \right]!\left( n-r \right)!} \\
 & =\frac{n!}{\left( n-n+r \right)!\left( n-r \right)!} \\
 & =\frac{n!}{r!\left( n-r \right)!}=\frac{n!}{\left( n-r \right)!r!}{{=}^{n}}{{C}_{r}}=LHS \\
\end{align}$
Hence statement 2 is true.
Also
Statement 2 is not a correct explanation for statement 1
Answer option is (D)
Additional information:
Statement 1 is from the topic number theory that deals with power of unit digit in the number while statement 2 is called deals with combination. Whereas this particular statement 2 is a rule for combination.

Note: In this question, we should firstly check if the statements are correct or not then we find that is there any relation between them. Finally we see which of the four options best suits as.