Statement – 1 : If $ƒ:\{1,2,3,4,5\}\to \{1,2,3,4,5\}$ then the number of onto functions such that $ƒ\left(i\right)\ne i$ is 42 
Statement – 2 : If n things are arranged in row, the number of ways in which they can be de- arranged so that no one of them occupies its original place is $n!(1-\frac{1}{1!}+\frac{1}{2!}-\mathrm{......}{(-1)}^n\frac{1}{n!})$ 
Statement – 3 : Number of ways of distribution of 12 identical balls into 3 identical boxes is 19
Statement – 4 : Number of ways of distribution of n identical objects among r persons, each one of whom can receive any number of objects is ${}^{n+r-1}C_{r-1}$ 

Which of the following must be truth value of above statements in that order.

The Number of onto funtions from the set having 5 elements to it self is $\begin{array}{rcl}{5}^5-5\left(4^5\right)+10\left(3^5\right)-10\left(2^5\right)+5\left(1^5\right)& =& 3125-5\left(1024\right)+10\left(243\right)-10\left(32\right)+5\\ & =& 120\end{array}$

but in that $f\left(x\right)\ne x$, it means dearrangement

$5!\left(1-1+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}\right)=60-20+5-1=44$

The statement 1 is false statement. 

Statement 2: If n things are arranged in row, the number of ways in which they can be de- arranged so that no one of them occupies its original place is $n!(1-\frac{1}{1!}+\frac{1}{2!}-\mathrm{......}{(-1)}^n\frac{1}{n!})$ 

This is formula, and correct

Statement 3: Number of ways of distribution of 12 identical balls into 3 identical boxes is 19

Yes this is correct. The possibiliites are $$\begin{gathered} \left(0,0,12\right),\left(0,1,11\right),\left(0,2,10\right),\left(0,3,9\right),\left(0,4,8\right),\left(0,5,7\right),\left(0,6,6\right),\left(1,1,10\right),\left(1,2,9\right) \\ \left(1,3,8\right),\left(1,4,7\right),\left(1,5,6\right),\left(2,3,7\right),\left(2,4,6\right),\left(2,5,5\right),\left(3,3,6\right),\left(3,4,5\right),\left(4,4,4\right) \end{gathered}$$

Statement 4: Number of ways of distribution of n identical objects among r persons, each one of whom can receive any number of objects is ${}^{n+r-1}C_{r-1}$

This is correct statement