Binomial Theorem: Evaluating Complex Combinations

In summary, the conversation discusses how to evaluate a series with changing lower limits in the combination part. The suggestion is to look at specific examples, such as when m=1 and m=2, to find a pattern and come up with a solution.
  • #1
ritwik06
580
0

Homework Statement



Evaluate
[tex]\sum^{m}_{r=0} ^{ n + r }C_{n}[/tex]

I can handle things when the lower thing in the combination part is changing, what shall I do with this one?
 
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  • #2
Try writing out a few terms in the series and see if it helps.
 
  • #3
I get this:
[tex]^{n}C_{n} + ^{n+1}C_{n} + ^{n+2}C_{n} + ... + ^{n+r}C_{n}
[/tex]
[tex]^{n}C_{0} + ^{n+1}C_{1} + ^{n+2}C_{2} + ... + ^{n+r}C_{r}
[/tex]

All I can do is this, now both the superscript and th subscript are increasing in A.P.
 
  • #4
The thread is still unsolved...
 
  • #5
the suggestion was that you actually look at a few specific examples.
If m= 1, you have
[tex]^nC_n+ ^{n+1}C_n= \frac{n!}{n!0!}+ \frac{(n+1)!}{n!1!}= 1+ n+ 1= n+ 2[/tex]
If m= 2, you have
[tex]^nC_n+ ^{n+1}C_n+ ^{n+2}C_n= n+ 2+ \frac{(n+2)!}{n! 2!}= n+ 2+ (n+1)(n+2)/2= n+2+ \frac{1}{2}n^2+ \frac{3}{2}x+ 1= \frac{1}{2}n^2+ \frac{5}{2}n+ 3[/tex]

Try a few more like that and see if anything comes to mind.
 

1. What is the Binomial Theorem?

The Binomial Theorem is a mathematical formula used to expand a binomial expression raised to a power. It allows for the efficient calculation of complex combinations without having to multiply out each term individually.

2. How do you evaluate complex combinations using the Binomial Theorem?

To evaluate complex combinations using the Binomial Theorem, you first need to identify the values of n (the power) and r (the term you are looking for) in the binomial expression. Then, you can plug these values into the formula (nCr) * a^(n-r) * b^r, where a and b represent the two terms in the binomial expression. Finally, you can simplify the expression to get the final answer.

3. What is the difference between a combination and a permutation in the context of the Binomial Theorem?

In the context of the Binomial Theorem, a combination refers to the number of ways to choose a certain number of objects from a larger set, regardless of their order. A permutation, on the other hand, refers to the number of ways to arrange a certain number of objects in a specific order. The Binomial Theorem is used to evaluate complex combinations, not permutations.

4. Can the Binomial Theorem be used for negative or fractional powers?

No, the Binomial Theorem can only be used for positive integer powers. This is because the formula (nCr) * a^(n-r) * b^r relies on the concept of combinations, which only applies to whole numbers. Negative or fractional powers would require the use of a different formula or method.

5. How is the Binomial Theorem used in real-world applications?

The Binomial Theorem has many applications in fields such as statistics, engineering, and physics. It can be used to calculate probabilities, determine coefficients in polynomial expansions, and solve equations involving combinations. It is also used in financial and economic models to predict future outcomes based on probabilities and combinations.

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