Binomial theorem

Question 1

Factorials eg 6!

Answer 1

6x5x4x3x2x1

Question 2

General formula for factorials

Answer 2

n(n-1)(n-2)…..* 3* 2 * 1 =n!

Question 3

0!=

Answer 3

1

Question 4

6! Can equal

Answer 4

6*5!

Question 5

Permutations

Answer 5

Random selection of objects when the order is important. e.g. different ways athletes can finish a race.

Question 6

^nPr=

Answer 6

n!/(n-r)!
Where n is the pool of objects and r is the number of objects arranged

Question 7

Combination definition

Answer 7

Random selection from a set of objects where the order is unimportant.
E.g. how many combinations of athletes qualifying to next round.

Question 8

^nCr more commonly written as

Answer 8

(n/r) *no /
n choose r
Where n is number of objects total and r is number chosen.

Question 9

(n/r) =
No /

Answer 9

n!/r!(n-r)!

Question 10

nCn-r =

Answer 10

nCr

Question 11

nCr-1 +nCr =

Answer 11

n+1Cr

Question 12

Pascal’s triangle

Answer 12

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.

Question 13

(a+b)^5 =

Answer 13

a^5+ 5a^4b + 10a^3b^2 +10a^2b^3 + 5ab^4 +b^5

Question 14

General formula for binomial theorem

Answer 14

(a+b)^n =[nEr=o] nCr a^(n-r)b^r

Question 15

What may we need to do with bracket that is to high power to get it to work for binomial theorem

Answer 15

Transform it so it fits the form (a+b)^n

Question 16

What does binomial theorem allow us to do?

Answer 16

Allows us to easily multiply out brackets to high powers,

Question 17

Binomial theorem process

Answer 17

Transform bracket to form way it is in equation, then carry out equation. Group like terms and Bob is your uncle.

Question 18

Binomial expansion involving 3 terms in the bracket.

Answer 18

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.

Question 19

Finding particular term in binomial expansion

Answer 19

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.