Hi! I am Aalam and i am going to be talking about Combinations and Binomial Theorem.
Combination:
- An unordered collection of elements.
- In permutation we select and order elements. But with a combination we only select the element.
- Must use the formula. Dash Method cannot be used.
- N = The total number of items in the sample.
- R = The number of items to be selected from in the sample.
- N > R
Example: A class consists of 12 girls and 10 boys. A committee is to be selected consisting of 7 members. In how many ways can this be done if:
A) There are to be 4 boys and 4 girls on the committee?
4 Boys: n = 10 Total number of boys.
r = 4 Number of boys that could be selected.
10C4 = 210
3 Girls: n = 12 Total number of girls.
r = 3 Number of girls that could be selected.
12C4 = 220
Total: 210x220 = 46200
B) There are to be at least 5 men on the committee?
5 Boys: n = 10 Total number of boys.
r = 5 At least 5 boys need to be selected.
10C5 = 252
2 Girls: n = 12 Total number of girls.
r = 2 As there are 5 boys and the committee consists of 7 people 2 girls could be selected.
12C2 = 66
Total: 252x66 = 16632
OR
6 Boys: n = 10 Total number of boys.
r = 6 6 boys could be selected as the question says at least 5 boys.
10C6 = 210
1 Girls: n = 12 Total number of girls.
r = 2 As there are 6 boys and the committee consists of 7 people 1 girls can be selected.
12C1 = 12
Total: 210x12 = 2520
OR
7 Boys n = 10 Total number of boys.
r = 7 7 boys could be selected as the question says at least 5 boys.
10C7 = 120
0 Girls as the committee consists of 7 people and we already got 7 boys.
Total = 16632+2520+120 = 19272
C) It does not matter which sex is chosen?
n = 22 As there are 10 boys and 22 girls and sex doesnt matter there are a total of 22 people.
r = 7 Only 7 can be selected.
22C7 = 170544
The Binomial Theorem:
(a+b)n = (nC0)(anb0) + (nC1)(an-1)(b1) + (nC2)(an-2)(b2) + … (nCn)( a0)(bn)
(a+b)n = (nC0)(anb0) + (nC1)(an-1)(b1) + (nC2)(an-2)(b2) + … (nCn)( a0)(bn)
- A binomial expansion where the exponent is n will have n+1 terms when expanded.
- A binomial expansion where the exponent is even will have an odd number of terms and will have a middle term when expanded. Ex. (x+y)^2
- A binomial expansion where the exponent is odd will have an even number of terms and will not have a middle term when expanded. Ex. (x+y)^3
Example: Expand and Simplify using the binomial expansion: (3x-y)5
a = 3x
b = -1y
n = 5
(3x-y)5 =
5C0 (3x)5(-y)0 + 5C1
(3x)4(-y)1 + 5C2 (3x)3(-y)2
+ 5C3 (3x)2(-y)3 + 5C4
(3x)1(-y)4 + 5C5 (3x)0(-y)5
= 1.243x5.1 + 5.81x4(-y)
+ 10. 27x3y2 + 10.9x2(-y)3 + 5.3xy4
+ 1. 1(-y)5
= 243x5
– 405x4y + 270x3y3 – 90x2y3
+ 15xy4 – y5
I hope you guys enjoyed this blog post have a great day!
I Choose!!
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