Binomial theorem Flashcards
Factorials eg 6!
6x5x4x3x2x1
General formula for factorials
n(n-1)(n-2)…..* 3* 2 * 1 =n!
0!=
1
6! Can equal
6*5!
Permutations
Random selection of objects when the order is important. e.g. different ways athletes can finish a race.
^nPr=
n!/(n-r)!
Where n is the pool of objects and r is the number of objects arranged
Combination definition
Random selection from a set of objects where the order is unimportant.
E.g. how many combinations of athletes qualifying to next round.
^nCr more commonly written as
(n/r) *no /
n choose r
Where n is number of objects total and r is number chosen.
(n/r) =
No /
n!/r!(n-r)!
nCn-r =
nCr
nCr-1 +nCr =
n+1Cr
Pascal’s triangle
Starting at (1+x)^0 we multiply out the brackets, for each level we go down in the triangle we increase the value of the index by one.
Coefficients will equal numbers of the triangle.
It also corresponds to answers to combinations.
(a+b)^5 =
a^5+ 5a^4b + 10a^3b^2 +10a^2b^3 + 5ab^4 +b^5
General formula for binomial theorem
(a+b)^n =[nEr=o] nCr a^(n-r)b^r
What may we need to do with bracket that is to high power to get it to work for binomial theorem
Transform it so it fits the form (a+b)^n
What does binomial theorem allow us to do?
Allows us to easily multiply out brackets to high powers,
Binomial theorem process
Transform bracket to form way it is in equation, then carry out equation. Group like terms and Bob is your uncle.
Binomial expansion involving 3 terms in the bracket.
Split into way it is in formula with one of the terms being multiple. Usually question will only be up to a number of terms, so add ….. if terms go past this.
Then binomially expand two terms when required, and solve rest of equation, group like terms up to certain powers.
Finding particular term in binomial expansion
Use general term with our known value for n and r. Rearrange so we have one particular term with r values. Work out r values and solve.
