Question

Evaluate the expression
5
Σ [5]
[K]
K=0 The 5 is over the K, but there is not a division sign, they are inside 1 pair of brackets together.

Answer 1

\[\sum_{k=0}^{5}\rightarrow \left(\begin{matrix}5 \\ k\end{matrix}\right)\] is what it looks like but with brackets and not perenthesis.

Answer 2

\[\left(\begin{matrix}5 \\ 0\end{matrix}\right)+\left(\begin{matrix}5 \\1\end{matrix}\right)+\left(\begin{matrix}5 \\2\end{matrix}\right)+\left(\begin{matrix}5 \\3\end{matrix}\right)+\left(\begin{matrix}5 \\ 4\end{matrix}\right)+\left(\begin{matrix}5 \\5\end{matrix}\right)\]
\[=\left(\begin{matrix}5+5+5+5+5+5 \\0+1+2+3+4+5\end{matrix}\right)=\left(\begin{matrix}30 \\15\end{matrix}\right)\]

Answer 3

What is the point of it being on the bottom in the brackets and under it though if it is not division or anything? Can you explain it a bit more?

Answer 4

is this supposed to be \(\binom{n}{k}\) the binomial aka "n choose k"

Answer 5

looks like calculating in combination formula.
do you know factorial ?
it like :
n! = n(n-1)(n-2)(n-3)... 3.2.1

nCr = n!/(r! * (n-r)!)

(5 0) --> n = 5 , r = 0
5C0 = 5!/(0! * (5-0)!) = 5! / 0!5! = 1

(5 1) --> n = 5 , r = 1
5C1 = 5!/(1! * (5-1)!) = 5! / 1!4! = 5

...
...
and so on tiil (5 5)
continue it

Answer 6

if so, answer is \(2^n\) or in your case \(2^5\)

Answer 7

\[\sum_{k=0}^n\binom{n}{k}=2^n\] always

Answer 8

you can see this if you add up any row in pascal's triangle, or you can prove it by writing
\[2^n=(1+1)^n=\sum_{k=0}^n\binom{n}{k}\]

Answer 9

but if that is not clear, we can just add them one by one and see that you get 32

Answer 10

Oh thank you for the reply! It's the first time I have seen a problem like this and I'm not used to it. But your explanation helped!

Answer 11

My teacher weirdly put it on the test and for some reason we never went over an example or anything like it!

Answer 12

do you know how to compute \(\binom{5}{0}\) and \(\binom{5}{1}\) etc?

Answer 13

Yes I believe so. I just never saw it in sigma notation like that.

Answer 14

that just means you are adding them up, add across a row in pascal's triangle

Answer 15

Alright, thanks for the help! :D

Answer 16

yw

Answer 17

if we take it as matrix then i have solved and if it is a binomial co-efficient then it is solved above.