Permutations and Combinations

Factors of 36 = 2².3².1

Five-digit combinations can be 

(1, 2, 3, 3), (1, 4, 3, 1), (1, 9, 2, 1), (1, 4, 9, 11), (1, 2, 3, 6, 1), (1, 6, 1, 1)

i.e., total numbers  $\frac{5!5!}{2!2!}+\frac{5!}{2!2!}+\frac{5!}{2!2!}+\frac{5!}{3!}+\frac{5!}{2!}+\frac{5!}{3!2!}=\left(30*3\right)+20+60+10=180.$  

EHINTZ

words starting with E is 5!

words starting with H is 5!

words starting with I is 5!

words starting with N is 5!

words starting with T is 5!

$9*9*3=81*3=243$

Single digit factor of $12\to \mathrm{1,2},\mathrm{3,4},6$

  $(x,y,z)$  where , $xyz=12$

$(\mathrm{6,2},1)(\mathrm{4,3},1)(\mathrm{3,2},2)$

$\begin{array}{ccc}\downarrow & \downarrow & \downarrow \\ \text{6Ways}& \text{6 Ways}& \text{3Ways}\end{array}$

 Total 15 Ways

(6!/ (5!1!) * 2! + (6!/ (4!2!) * 2! + (6!/ (3!)²2!) * 2! = 62

All even + 2 odd 1 even
¹? C? + ¹? C? * ¹? C?
(10*9*8)/6 + (10*9)/2 * 10
120 + 450 = 570

$x_1+x_2=6,13\to 6+6=12$

$\begin{array}{l}{x}_1+x_2=4,11,18\to 4+8+1=13\\ x_1+x_2=2,9,16\to 2+9+3=14\\ x_1+x_2=7,14\to 7+5=12\\ x_1+x_2=5,12\to 5+7=12\end{array}$

= 63

I (6)       F (8)

Case I       2            4

Case II      3            6

Case III     4            8

Total =   

All even + 2 odd 1 even
¹? C? + ¹? C? * ¹? C?
120 + 450
=570

There are 7 letters permutation.
The total number of 4 letters
Out of 2A's, 2K's and 3 different letters O, L, T.
= Coefficient of x? in 4! (1 + x + x²/2!)² (1 + x)³
= Coefficient of x? in 4! (1 + x + x²/2)² (1 + x)³
= Coefficient of x? in 4! [ (1 + x)? + (1 + x)? x² + (1 + x)³ + x? /4]
= 4! [? C? +? C? + ³C? /4] = 4! (5 + 6 + 1/4)
= 24 [11 + 1/4]
= 264 + 6 = 270