Number of squarefree positive integers less than $100$ 
An integer is called squarefree if it is not divisible by the square
  of a positive integer greater than $1$. Find the number of squarefree
  positive integers less than $100$.

My attempt:
I apply the inclusion-exclusion principle directly.


*

*Total number of integers = 99

*Number of integers divisible by $2^2$ = $24$

*Number of integers divisible by $3^2$ = $11$

*Number of integers divisible by $4^2$ = $6$

*Number of integers divisible by $5^2$ = $3$

*Number of integers divisible by $6^2$ = $2$

*Number of integers divisible by $7^2$ = $2$

*Number of integers divisible by $8^2$ = $1$

*Number of integers divisible by $9^2$ = $1$

*Number of integers divisible by $2^2$ and $3^2$ = $2$

*Number of integers divisible by $2^2$ and $4^2$ = $1$


Then the required solution would be $99-(24+11+6+3+2+2+1+1)+2+1=52$. But the solution is $61$. Where is my mistake?

If $4^2$ divides $n$ then so does $2^2$, so your method double-counts. A number is nonsquarefree iff it is divisible by a square of a prime, so we can proceed as follows: 
Let $A_k$ denote the numbers (here, positive integers) less than $100$ divisible by $k^2$. Since $$ |A_k| = \left\lfloor \frac{100 - 1}{k^2} \right\rfloor ,$$ we have $|A_k| = 0$ for $k \geq 10$. Then, applying the inclusion-exclusion principle to the sets $A_p$, $p < 10$ prime, gives that the number of nonsquarefree numbers less than $100$ is
\begin{multline}|A_2| + |A_3| + |A_5| + |A_7| \\- (|A_2 \cap A_3| + |A_2 \cap A_5| + |A_2 \cap A_7| + |A_3 \cap A_5| + |A_3 \cap A_7| + |A_5 \cap A_7|) + \cdots ,\end{multline} where $\cdots$ denotes terms containing cardinalities of intersections of three and four sets.
Now, $n \in A_q \cap A_{q'}$ for $q, q'$ coprime iff $n$ is divisible by both $q^2$ and $(q')^2$, and hence by coprimality, by $q^2 (q')^2 = (qq')^2$, so $A_q \cap A_{q'} = A_{qq'}$. So, for example, $A_2 \cap A_5 = A_{10}$, which by the above is empty, and the same is true for all of the intersections of two sets except $A_2 \cap A_3 = A_6$. Similarly, by induction, the triple and quadruple intersections are all empty. This leaves that the number of nonsquarefree numbers less than $100$ is
$$|A_2| + |A_3| + |A_5| + |A_7| - |A_6| .$$
