Simplify \sum_{i=0}^{n}\binom{n}{i}2^{n-i} to 3^n

  • Thread starter noblerare
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In summary, the conversation is about simplifying an equation to prove that the number of ways of counting something is equal to 3^n. The conversation includes an equation and a hint to help simplify it.
  • #1
noblerare
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Homework Statement



This is actually part of a larger problem that asks us to prove that the number of ways of counting something is equal to [tex]3^n[/tex]. I have worked it out and the equation I get is:

[tex]\binom{n}{0}2^n + \binom{n}{1}2^{n-1}+\ldots+\binom{n}{n}2^{n-n}[/tex]

I am wondering how I should simplify this to make it equal to [tex]3^n[/tex]

2. The attempt at a solution

I rewrote the above equation into:

[tex]\displaystyle\sum_{i=0}^{n}\binom{n}{i}2^{n-i}[/tex]

But then I didn't know how to proceed from here since both the combinatorial choosing term and the powered terms are changing. I also tried factoring out [tex]2^n[/tex] but that didn't do anything.

Can anyone help me?
Thanks.
 
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  • #2
Hi noblerare! :smile:

Hint: what's (2 + x)n ? :wink:
 
  • #3
Ohhhh, wow. Okay thanks, tiny-tim! Problem solved.
 

1. How do you simplify the expression i=0n &binom;ni 2n-i to 3n?

To simplify this expression, we can use the binomial theorem. First, we expand the binomial coefficient using the formula &binom;ni = n! / (i! (n-i)!). This gives us an expression of i=0n n! / (i! (n-i)!) 2n-i. We can then rearrange this expression to n! 2n i=0n 1 / (i! (n-i)!). Finally, we use the fact that i=0n &binom;ni xi = (1+x)n to substitute 3 for x, giving us n! 2n (1+3)n, which simplifies to 3n n! 2n.

2. What is the binomial theorem and how is it used to simplify this expression?

The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial. In this case, the binomial is (1+x)n, and the formula states that i=0n &binom;ni xi = (1+x)n. This theorem allows us to simplify the expression i=0n &binom;ni 2n-i to 3n by substituting 3 for x.

3. Can this expression be simplified further?

Yes, the expression can be simplified further by using the fact that n! = n (n-1) (n-2) ... 1. This gives us 3n n (n-1) (n-2) ... 1 2n, which can be written as 3n n! 2n.

4. What is the significance of this expression in mathematics?

This expression is significant in combinatorics, the branch of mathematics that studies the counting and arrangement of objects. It represents the number of ways to choose a subset of n objects from a set of 2^n objects, and is therefore useful in solving problems involving combinations and permutations.

5. Can this expression be applied to real-life situations?

Yes, this expression can be applied to real-life situations through its application in combinatorics. For example, it can be used to calculate the number of possible combinations in a lottery or the number of possible seating arrangements at a dinner table. It can also be used in computer science and coding to calculate the number of possible outcomes in a program or algorithm.

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