Pascal Triangle
I know that (x+y)^5 is
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
what does (x+y+2)^5
mean as a pascal triangle?
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
what does (x+y+2)^5
mean as a pascal triangle?
Pascal triangle is for powers of binomials. That's a trinomial, so you group it as a binomial.
( (x) + (y+2) )^5.
Now you can use your co-efficients like you would with a binomial. Then you could further simplify the powers of (y+2) that you get similarly with pascal's triangle (if they've asked you to).
For calculating the co-efficients you can use the formula.
https://en.wikipedia.org/wiki/Pascal%27s_triangle
By the way this isn't a programming question, granted that you might have gotten that as a programming assignment.
( (x) + (y+2) )^5.
Now you can use your co-efficients like you would with a binomial. Then you could further simplify the powers of (y+2) that you get similarly with pascal's triangle (if they've asked you to).
For calculating the co-efficients you can use the formula.
https://en.wikipedia.org/wiki/Pascal%27s_triangle
By the way this isn't a programming question, granted that you might have gotten that as a programming assignment.
Google "Pascal's pyramid".
Alternatively, Pascal's triangle can be used as a "look-up table" for binomial coefficients, nCr, where nCr is the number of ways of choosing r objects from n. Specifically, nCr is the [r] member of the [n] row(where, here, square brackets signify counting from 0).
If you had the trinomial expansion of ( x + y + z )n then, by picking one term from each bracket in multiplying out, the coefficient of the term
xaybzc (where a+b+c=n)
would be
nCa.(n-a)Cb
which is a product of two terms in Pascal's triangle. (It is also given by n!/(a!b!c!) )
Alternatively, Pascal's triangle can be used as a "look-up table" for binomial coefficients, nCr, where nCr is the number of ways of choosing r objects from n. Specifically, nCr is the [r] member of the [n] row(where, here, square brackets signify counting from 0).
If you had the trinomial expansion of ( x + y + z )n then, by picking one term from each bracket in multiplying out, the coefficient of the term
xaybzc (where a+b+c=n)
would be
nCa.(n-a)Cb
which is a product of two terms in Pascal's triangle. (It is also given by n!/(a!b!c!) )
Last edited on
more useless info … until the coefficients move to double digits, so for most practical examples (when using it for coefficients), pascal's triangle is also the powers of 11.
11 to the 0 is 1
11 to the 1 is 11
11 to the 2 is 121
1331
etc :)
11 to the 0 is 1
11 to the 1 is 11
11 to the 2 is 121
1331
etc :)
Last edited on
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