Oriented circuit board problem

Question: In the layout of an oriented circuit board for an electronic product, there are 15 different locations for chips.

(a)  If 10 different types of chips are to be placed on the board, how many different layouts are possible?

(b)   If the 10 chips that are placed on the board are the same type, how many different layouts are possible?

(c)     If the 10 chips that are placed on the board are divided in 6 chips of the same type (Type I) and 4 chips of another type (Type II), how many different layouts are possible?

Answer:

(a)    Since, all the 10 chips are different type, they can be placed on a board of 15 locations in ₁₅P₁₀ ways

Equivalently, first chip can be placed in any of the 15 locations, second chip can be placed in any of the remaining 14 locations and so on… i.e. 15*14*13*12*11*10*9*8*7*6 = 10897286400

So, if 10 different types of chips are to be placed on the board with 15 locations, 10897286400 different layouts are possible.

(b)      For the same type of chips, there would be ₁₅C₁₀ ways

Those 10 chips can be placed the locations shown in first diagram, or in 2^nd diagram, or 3^rd or so on and there will such ₁₅C₁₀ combinations

So, if 10 chips that are placed on the board are the same type, 3003 different layouts are possible.

(c)    Now if there are 6 chips are of the same type (type I) and 4 are of another type (type II) they can be placed in

₁₅C₆*₁₅C₄ = 5005*1365 =  6831825

So, if the chips that are placed on the board are divided in 6 chips of the same type and 4 chips of another type then 6831825 different layouts are possible.