Calculating Sum of Binomial Coefficients in Terms of a and n

In summary, binomial coefficients are mathematical terms used to represent the coefficients in a binomial expansion and calculate the number of ways to choose objects from a larger set. They can be calculated using the formula n choose k, and are significant in probability for calculating the probability of a specific number of successes. Binomial coefficients are always positive integers and have various applications in real life, such as in probability, genetics, and computer science.
  • #1
ritwik06
580
0

Homework Statement


If [tex]\sum^{n}_{r=0} \frac{1}{^{n}C_{r}} = a[/tex], then find the value of [tex]\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}[/tex] in terms of a and n.[/tex]






The Attempt at a Solution


I tried to write down the terms of both the series, but to no avail. i can't think of anything.Please shed some light.
 
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  • #2
ritwik06 said:

Homework Statement


If [tex]\sum^{n}_{r=0} \frac{1}{^{n}C_{r}} = a[/tex], then find the value of [tex]\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}[/tex] in terms of a and n.[/tex]

Hi ritwik06! :smile:

Hint: suppose n = 12.

Then [tex]\sum^{n}_{r=0} \frac{1}{^{n}C_{r}}[/tex]

= (0!12! + 1!11! + 2!10! + 3!9! + …)/12!

So what is [tex]\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}[/tex] ? :smile:
 
  • #3
Hi tim, I'm not seeing how this helps to solve the problem. You have a term dependent r in each summand, so how do we express it in a?
 
  • #4
tiny-tim said:
Hi ritwik06! :smile:

Hint: suppose n = 12.

Then [tex]\sum^{n}_{r=0} \frac{1}{^{n}C_{r}}[/tex]

= (0!12! + 1!11! + 2!10! + 3!9! + …)/12!

So what is [tex]\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}[/tex] ? :smile:

Thank god! Somebody helped me. But Tim, I wonder what you wish to convey... Please could you be more explicit :smile:
 
  • #5
Consider:
[tex]
\sum^{n}_{r=0} \frac{n-r}{^{n}C_{r}}
[/tex]
How does that compare with:
[tex]
\sum^{n}_{r=0} \frac{r}{^{n}C_{r}}
[/tex]
Does that give you any ideas??
 
  • #6
Hi ritwik06! :smile:

Have you got this now … you haven't said?

If you haven't, then follow Dick's hint … it's much better than mine! :redface:

(same for the other thread)
 
  • #7
That's nice of you to say, tiny-tim. Thanks. :) Now you've got me curious. ritwik06, did you get it? It's surprising easy if you think about it right, and pretty nonobvious if you don't. It took me a while.
 

1. What are binomial coefficients?

Binomial coefficients are mathematical terms that represent the coefficients of the terms in a binomial expansion. They are used to calculate the number of ways to choose a specific number of objects from a larger set of objects.

2. How do you calculate binomial coefficients?

Binomial coefficients can be calculated using a formula, where n represents the total number of objects and k represents the number of objects chosen. The formula is n choose k, or nCk, and is equal to n! / (k!(n-k)!), where ! represents the factorial function.

3. What is the significance of binomial coefficients in probability?

Binomial coefficients are used in probability to calculate the probability of a specific number of successes in a series of independent events. They are also used in the binomial distribution, which represents the probability distribution of a binomial random variable.

4. Can binomial coefficients be negative?

No, binomial coefficients are always positive integers. This is because they represent the number of ways to choose objects, and negative numbers do not make sense in this context.

5. How are binomial coefficients used in real life?

Binomial coefficients have many real-life applications, including in probability and statistics, genetics, and computer science. They can be used to solve problems involving combinations and permutations, such as in card games or in choosing lottery numbers. They are also used in calculating probabilities in fields such as finance and physics.

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