Divide nCr-1 and nCr ( i.e. nCr-1/nCr =56/28).. So after solving we get the foll. Eqn. 3r=2n+2 .... Now similarly divide nCr/nCr+1 = 28 /8 ... So we get another eqn. That is 9r= 7n-2.. Now solving these two equations we will get r=6..

Solve 2 equations simultaneously.eqn 1: [n!/(r-1)!(n-(r-1))!]÷[n!/r!(n-r)! = 56/28eqn 2: [n!/(r+1)!(n-(r+1))!]÷[n!/r!(n-r)! = 8/28you'll get 2 eqns, solve them and you'll get the value of r.

Given:
nCr-1 = 56
nCr = 28
nCr+1 = 8

To find:
The value of r

Solution:
Let's start by understanding the concept of combinations and the formula for nCr.

In mathematics, combinations are ways to select items from a larger set without considering the order. The formula for combinations is given by:

nCr = n! / (r!(n-r)!)

Where n is the total number of items, r is the number of items to be selected, and ! denotes factorial, which means multiplying a number with all the positive integers less than it.

Now, let's solve the given problem step by step.

Step 1:
Given that nCr-1 = 56 and nCr = 28, we can write the following equations:

nCr-1 = n! / ((r-1)!(n-r+1)!) = 56
nCr = n! / (r!(n-r)!) = 28

Step 2:
To simplify the equations, we can divide the two equations:

(n! / ((r-1)!(n-r+1)!)) / (n! / (r!(n-r)!)) = 56 / 28

Simplifying further, we get:

(r!(n-r)!)/( (r-1)!(n-r+1)! ) = 2

Step 3:
Now, let's simplify the factorial terms:

(r!(n-r)!)/( (r-1)!(n-r+1)! ) = 2
(r!(n-r)!)/( (r-1)!(n-r+1)(n-r)! ) = 2
(r!)/( (r-1)!(n-r+1) ) = 2

Step 4:
Expanding the factorial terms:

(r(r-1)!)/( (r-1)!(n-r+1) ) = 2

Simplifying further, we get:

r = 2(n-r+1)

Step 5:
Given that nCr+1 = 8, we can write the equation:

nCr+1 = n! / ((r+1)!(n-r-1)!) = 8

Expanding the factorial terms:

n! / ((r+1)!(n-r-1)!) = 8

Step 6:
Substituting the value of nCr from the previous equation:

28 / ((r+1)!((n-r)!)(n-r-1)!) = 8

Simplifying further, we get:

(r+1)!((n-r)!)(n-r-1)! = 28/8 = 7/2

Step 7:
Since r is a positive integer, the only possible value for (r+1)! is 3.

Therefore, (r+1)! = 3

Step 8:
Solving for r:

r+1 = 3
r = 3 - 1
r = 2

Answer:
The value of r is 2.