Math Help! (C(6,0))^2 + (C(6,1))^2 + (C(6,2))^2 + (C(6,3))^2 + (C(6,4))^2 + (C(6,5))^2 +(C(6,6))^2 = C(k,w)
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HOME > > Math Help! (C(6,0))^2 + (C(6,1))^2 + (C(6,2))^2 + (C(6,3))^2 + (C(6,4))^2 + (C(6,5))^2 +(C(6,6))^2 = C(k,w)

Math Help! (C(6,0))^2 + (C(6,1))^2 + (C(6,2))^2 + (C(6,3))^2 + (C(6,4))^2 + (C(6,5))^2 +(C(6,6))^2 = C(k,w)

[From: ] [author: ] [Date: 12-02-21] [Hit: ]
C(6,0) = [1]C(6,1) = [6]... [15] .......
where k and w are positive integers. Find the smallest possible value of (k+w)

Please explain. Thanks!

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Pascals triangle can be written in combination form. (see the link)

C(6,0) = [1] C(6,1) = [6] ... [15] ... [20] ... [15] ... [6] ... [1]

Then this sum is 1+36+225+400+225+36+1 = 924 = C(12,7) by Pascal's triangle.
So the value of k+w here is 19

Then we need to look and see if there is any k+w less than 19.

In Pascal's triangle, what that means is you need to look at row 19 as this is the max value of both k and w. Any cells in that row, or lesser rows, that are the same sum 924 are possibilities.
Notice that 924 is the max in row 12, so we need to examine cells in row 13-19.

None exist, so 19 is the smallest possible value for k+w.

I don't know of a different way to do this. It may be out there, but this works, and is fairly simple if you have a triangle out to row 19 that you can examine.

I hope this helps.
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