Understanding Binomial Coefficients: Finding the r-th term formula

In summary, the conversation is about verifying the formula for the general term in an IB mathematics portfolio. The formula is T_n+1=C_n^r a^{n-r} b^r and the conversation discusses using examples to verify the formula and figuring out the formula for the r-th term. It also mentions the use of Pascal's triangle and asks for clarification on the variables a and b. There is also mention of expanding (a+b)^n and a request for help with the (n-1) term.
  • #1
gschjetne
95
0
I'm working on an IB mathematics portfolio, and here's a problem I don't understand:
The formula for the general term is
[tex]
T_n+1=C_n^r a^{n-r} b^r
[/tex]
Verify this formula by examples. This is the formula for the [itex](r+1)[/itex]-th term.
What would the formula for the [itex]r[/itex]-th term be?
It's not specified what [itex]a[/itex] and [itex]b[/itex] is supposed to represent. That's where I need some explanation.
I already made a nice Pascal triangle all the way to the 10th row.
All help is appreciated.
 
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  • #2
My guess is that it comes from the expansion:

[tex] (a+b)^{n}=\sum_{k=0}^{n}C_{n}^{k} a^{n-k}b^{k} [/tex]

Daniel.

PS.Can u handle the [itex] (n-1) [/itex] term part??
 
  • #3
Thanks, I'll give it a try
 

FAQ: Understanding Binomial Coefficients: Finding the r-th term formula

What is a binomial coefficient?

A binomial coefficient is a mathematical term that represents the number of ways to choose a subset of objects from a larger set. It is denoted by the expression (n choose k), where n is the total number of objects and k is the number of objects in the subset.

How do you calculate a binomial coefficient?

The binomial coefficient can be calculated using the formula (n choose k) = n! / (k!(n-k)!), where n! (read as "n factorial") represents the product of all positive integers less than or equal to n. Alternatively, you can also use the Pascal's triangle or a combination calculator to determine the binomial coefficient.

What is the significance of binomial coefficients?

Binomial coefficients have many practical applications in mathematics, statistics, and computer science. They are used to calculate probabilities in various scenarios, such as coin tosses, card draws, and genetics. They are also used in the binomial theorem, which is used to expand expressions of the form (a+b)^n.

Can binomial coefficients be negative?

No, binomial coefficients cannot be negative. They represent the number of ways to choose a subset of objects, so they must always be positive or zero. If the result of the calculation is negative, it is considered to be equal to zero.

What are some common misconceptions about binomial coefficients?

One common misconception is that binomial coefficients only apply to binomial expressions (a+b)^n. In reality, they can be used for any type of combination, including (a+b+c)^n or (a+b+c+d)^n. Another misconception is that binomial coefficients are only relevant to theoretical mathematics, when in fact they have many practical applications as mentioned earlier.

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