1.2. Binomial theorem

Question 1

Pascal’s triangle

Answer 1

any number in the triangle is the sum of the two numbers directly above it

Question 2

the number of combinations of n items taken r at a time is found by:

Answer 2

(n)
(r) - binomial coefficient (“n choose r”)
= C^nˇr =nˇCˇr = n^Cˇr
(n) n!
(r) = r!(n-r)!

Question 3

n! =

Answer 3

123…(n-1)*n (n factorial)
n>=1 (0!=1)

Question 4

C^nˇr

Answer 4

n -1=row in Pascal’s triangle
r-1=place in the specific row

Question 5

exponents of the first term always ___ by one
exponents of the second term always ___ by one
coefficients are determined by…

Answer 5

decrease, increase, the row of Pascal’s triangle the binomial looks like

Question 6

the binomial theorem

Answer 6

(n) (n) (n)
(a+b)^n = (0) a^n + (1) a^(n-1)b…(n) b^n
((n/r) is the binomial coefficient of a^(n-r) b^r, and r=0,1,2,3…)

Question 7

the general term ((r+1)th term):

Answer 7

(n)
Tˇ(r+1)=(r) a^(n-r) b^r

Question 8

binomial theorem written using sigma notation:

Answer 8

n
(Sigma) (n/r) a^(n-r) b^r
(r=0)

Question 9

mathematical proof

Answer 9

a series of logical steps that show that one side of a mathematical statement is equivalent to the other side for all values of variables - at the end of a proof we write a concluding statement, such as LHS=- RHS (two sides equal by definition) or QED