• calculate the sum of binomial coefficients as under :
• ∑(n-r+1)*C(n,m) within range r = m to n)

To calculate the sum of the binomial coefficients given by the expression ∑(n-r+1) * C(n, m) for r ranging from m to n, we need to break down the components of the expression and analyze how they interact. This involves understanding binomial coefficients and how they can be manipulated in summation.

Understanding Binomial Coefficients

The binomial coefficient C(n, m), also denoted as "n choose m," represents the number of ways to choose m elements from a set of n elements. It is calculated using the formula:

C(n, m) = n! / (m! * (n - m)!)

where "!" denotes factorial, the product of all positive integers up to that number.

Setting Up the Summation

In our case, we are summing the expression (n - r + 1) * C(n, m) as r varies from m to n. This means we will be evaluating the sum for each integer value of r in that range.

Rewriting the Sum

The sum can be expressed as:

• ∑(n - r + 1) * C(n, m) for r = m to n

We can rewrite (n - r + 1) as (n + 1 - r), which might help in simplifying the expression further.

Breaking Down the Summation

Now, let’s analyze the sum step by step:

• First, we can separate the summation into two parts:
• ∑(n + 1) * C(n, m) - ∑r * C(n, m)

Calculating Each Part

1. **First Part:** The first part, ∑(n + 1) * C(n, m), can be simplified since (n + 1) is a constant with respect to r. Thus, we can factor it out:

(n + 1) * ∑C(n, m) from r = m to n.

Since the sum of C(n, m) over all m from 0 to n is equal to 2^n, we can focus on the specific range of m to n.

2. **Second Part:** The second part involves summing r * C(n, m). This can be approached using the identity:

r * C(n, m) = n * C(n - 1, m - 1).

Thus, we can rewrite the second sum as:

∑n * C(n - 1, m - 1) from r = m to n.

This can also be simplified using properties of binomial coefficients.

Final Calculation

Putting it all together, we can derive the final expression for the sum:

∑(n - r + 1) * C(n, m) = (n + 1) * C(n + 1, m + 1) - n * C(n, m).

This gives us a clear formula to compute the sum of the binomial coefficients in the specified range.

Example Calculation

For a concrete example, let’s say n = 5 and m = 2:

• Calculate ∑(5 - r + 1) * C(5, 2) for r = 2 to 5.
• Evaluate each term: C(5, 2) = 10.
• Then compute the sum for r = 2, 3, 4, 5.

This process will yield the specific sum based on the values plugged into the derived formula.

In summary, the sum of the binomial coefficients can be computed systematically by breaking down the expression and applying known identities and properties of binomial coefficients. This approach not only simplifies the calculation but also enhances understanding of the underlying combinatorial principles.