Mathematics Dictionary
Dr. K. G. Shih
Polynomial coefficients and distribution
Symbol Defintion
Example : Sqr(x) = square root of x
Q01 |
- Coefficients of expansion of (a + b + c)^n
Q02 |
- Coefficients of expansion of (a + b + c)^2
Q03 |
- Coefficients of expansion of (a + b + c)^3
Q04 |
- Coefficients of expansion of (a + b + c)^3
Q05 |
- References
Q01. Coefficients in expansion of (a + b + c)^2
Coefficnets of (a^p)*(b^q)*(c^R) in expansion of (a + b + c)^n
p + q + r = n
Coefficients = (n!)/((p!)*(q!)*(r!))
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Q02. Cefficients of expansion of (a + b + c)^2
Formula
(a + b + c)^2 = a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*c*a
It has 6 terms
Coefficient formula
Coefficients of (a^p)*(b^q)*(c^R) in expansion of (a + b + c)^n
It is D = (n!)/((p!)*(q!)*(r!))
Where p + q + r = n
Example : Find coefficient a*b in expansion of (a + b + c)^2
n = 2, p = 1, q = 1 and r = 0
D = (2!)/((1!)*(1!)*(0!) = 2/(1*1*1) = 2
Note
0! = 1
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Q03. Cefficients of expansion of (a + b + c)^3
Formula of (a + b + c)^3
(a + b + c)^2 = a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*c*a
(a + b + c)^3 = ((a + b + c)^2)*(a + b + c)
a*(a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*c*a)
= a^3 + a*b^2 + a*c^2 + 2*(a^2)*b + 2*a*b*c + 2*c*(a^2)
b*(a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*c*a)
= b*a^2 + b^3 + b*c^2 + 2*a*b^2 + 2*(b^2)*c + 2*a*b*c
c*(a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*c*a)
= c*a^2 + c*b^2 + c^3 + 2*a*b*c + 2*b*(c^2) + 2*(c^2)*a
Hence (a + b + c)^3 = a^3 + b^3 + c^3
+ a*(b^2) + a*(c^2) + 2*(a^2)*b + 2*c*(a^2)
+ b*(a^2) + b*(c^2) + 2*a*(b^2) + 2*(b^2)*c
+ c*(a^2) + c*(b^2) + 2*b*(c^2) + 2*(c^2)*a
+ 6*a*b*c
= a^3 + b^3 + c^3
+ 3*a*(b^2) + 3*a*(c^2) + 3*b*(a^2) + 3*b*(c^2) + 3*c*(a^2) + 3*c*(b^2)
+ 6*a*b*c
It has 10 terms
Find coefficient of a*b*c in expansion of (a + b + c)^3
n = 3, p = 1, q = 1, r = 1 and p + q + r = 3
Coeff of a*b*c is (3!)/((1!)*(1!)*(1!)) = (3*2*1)/(1*1*1) = 6
Find coefficient of (a^2)*b in expansion of (a + b + c)^3
n = 3, p = 2, q = 1, r = 0 and p + q + r = 3
Coeff of a*b*c is (3!)/((2!)*(1!)*(0!)) = (3*2*1)/(2*1*1) = 3
Find coefficient of (a^3) in expansion of (a + b + c)^3
n = 3, p = 3, q = 0, r = 0 and p + q + r = 3
Coeff of a*b*c is (3!)/((3!)*(0!)*(0!)) = (3*2*1)/(3*2*1*1) = 1
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Q04. Cefficients of expansion of (a + b + c)^4
How many terms in the expansion of (a + b + c)^4 ?
(a + b + c)^0 has 1 term
(a + b + c)^1 has 3 terms
(a + b + c)^2 has 6 terms
(a + b + c)^3 has 10 terms
The number of terms are sequence of triangular pattern
Hence next number is 15
(a + b + c)^4 has 15 terms
What are the 15 terms in expansion of (a + b + c)^4 ?
They are
a^4, b^4, c^4
(a^3)*b, (a^3)*c, (b^3)*c, (b^3)*a, (c^3)*a, (c^3)*b
(a^2)*(b^2), (b^2)*(c^2), (c^2)*(a^2)
(a^2)*b*c, (b^2)*c*a, (c^2)*a*b
It has 15 terms
Find coefficient of (a^2)*b*c in expansion of (a + b + c)^4
n = 4, p = 2, q = 1, r = 1 and p + q + r = 4
Coeff of (a^2)*b*c is (4!)/((2!)*(1!)*(1!)) = (4*3*2*1)/(2*1*1) = 14
Find coefficient of (a^2)*(b^2) in expansion of (a + b + c)^4
n = 4, p = 2, q = 2, r = 0 and p + q + r = 4
Coeff of a*b*c is (4!)/((2!)*(2!)*(0!)) = (4*3*2*1)/(2*2*1) = 6
Find coefficient of (a^3)*b in expansion of (a + b + c)^4
n = 4, p = 3, q = 1, r = 0 and p + q + r = 4
Coeff of a*b*c is (4!)/((3!)*(1!)*(0!)) = (4*3*2*1)/(3*2*1*1*1) = 4
Question
1. How many terms in the expansion of (a + b + c)^4 ? Answer 21 terms
2. How many terms in the expansion of (a + b + c)^m ? Answer (m + 1)*(m + 2)/2
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Q4. References
Reference :
Numbers
Program 14 and 15
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