Question

How to find the coefficient in binomial expansion using permutation combination of three variables?

Answer 1

These expressions exhibit many patterns:

Each expansion has one more term than the power on the binomial.

The sum of the exponents in each term in the expansion is the same as the power on the binomial.

The powers on a in the expansion decrease by 1 with each successive term, while the powers on b increase by 1.

The coefficients form a symmetrical pattern.

Each coefficient entry below the second row is the sum of the closest pair of numbers in the line directly above it.

This triangular array is called Pascal's triangle,named after the French mathematician Blaise Pascal.
Pascal's triangle can be extended to find the coefficients for raising a binomial to any whole number exponent. This same array could be expressed using the factorial symbol,

In general, 
The symbol  called the binomial coefficient, is
Therefore, 
This could be further condensed using sigma notation.

This formula is known as the binomial theorem.

Answer 2

Answer:

Step-by-step explanation:

suppose x+y+z=n

list out the possibilities in the power random variable c

suppose possibilities of x are 1,2 and 3 , y has 2,3 and 4 and z has 1,2 and 3

then write x equivalent as c+c^2+c^3

y equivalent as c^2+c^3+c^4

z equivalent as  c+c^2+c^3

multiply these expressions

you will get

(c+c^2+c^3)^2(c^2+c^3+c^4)

in above expression find coefficient of c^n

that is your answer