Pascal's Triangle and Binomial Thereom?

Question

what are the differences and similarities between Pascal's Triangle and the Bi-nomial Theroem

Answer 1

Pascals Triangle allows you to calculate the coefficients when multiplying out polynomials.

e.g. Pascals Triangle:

1

11

121

1331

14641

etc

and (x+1)^2 = x^2 + 2x +1 coefficients 121

(x+1)^3 = x^3 + 3x^2 + 3x +1 coefficients 1331

(x+1)^4 = x^4 + 4x^3 + 6x^2 + 4x +1 coefficients 14641



So the coefficients of (x+1)^a correspond to the (a+1)th row of Pascals Triangle.

Answer 2

both have to do with a series of choices.



For example tossing a coin



with binomial theory you could work out the probability of getting x heads when you toss the coin y times.



Constructing pascals triangle would tell you the number of possiblities which lead to 0, 1, 2 , 3 ... heads given y coin tosses, which could then be used to work out the probabilities of each outcome.



So both deal with similar systems and both can be used to compute probabilties however pascals triangle takes longer, finds all possibilities and I think it can not find the probabilities as easily if the system has uneven probabilities (e.g. 60% heads 40% tails)

Answer 3

Nick J has is right ( they're also powers of 11)