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Mathematics Dictionary
Dr. K. G. Shih
Pascal triangle and Polynomial expansion
Subjects


  • Q01 | - Pascal triangle and sequences
  • Q02 | - Find coefficient of (x^3)*(y^5) in expansion of (x + y)^2
  • Q03 | - Find seqquence of C(m+4, 5) in Pascal triangle
  • Q04 | - Number of terms in expansion of (a + b + c)^n
  • Q05 | - Number of terms in expansion of (a + b + c + d)^n
  • Q06 | - Number of terms in expansion of (a + b + c + d + e)^n
    • Answers


    Q01. Pascal triangle and equences 

    

   n  r = 0   1   2   3   4   5   6   7   8   9

   0      1
   1      1   1 
   2      1   2   1
   3      1   3   3   1
   4      1   4   6   4   1
   5      1   5  10  10   5   1
   6      1   6  15  20  15   6   1
   7      1   7  21  35  35  21   7   1
   8      1   8  28  56  70  56  28   8   1
   9      1   9  36  84 126 126  84  36   9   1
  10     10  45 120 210 252 210 120  45  10   1  




Pascal triangle

     1. It used to find coefficients of bionmial expansion
     2. It can be used to find number squence
     3. It can be used to find Fibonacci's seqquences
    Sequences
    • r = 2 : Sequence 1, 3, 6, 10, 15, ...
      • T(n) = n*(n + 1)/2
      • S(n) = n*(n + 1)*(n + 2)/(3!)
      • Sum[C(n + 1, 2)] = C(n + 2, 3)
      • It is triangular number sequence
    • r = 3 : Sequence 1, 4, 10, 20, 35, ...
      • T(n) = n*(n + 1)*(n + 2)/(3!)
      • S(n) = n*(n + 1)*(n + 2)*(n + 3)/(4!)
      • Sum[C(n + 2, 3)] = C(n + 3, 4)
      • The 1st difference is triangular number sequence

    Go to Begin

    Q02. Find coefficients of (x^3)*(y^5) in (x + y)^n

    Solution

   Since n is integer, hence n = 3 + 5 + 8
   Hence the coefficnet of (x^3)*(y^5) is 56

Verify

   Coefficient of (x^3)*(y^3) is C(8, 3)
   C(8, 3) = (8*7*6)/(3!) = 56

    Go to Begin

    Q03 Find seqquence of C(m+4, 5) in Pascal triangle

    Squecne of C(m + 4, 5)

     m = 1 : C(5, 5) = 1
     m = 2 : C(6, 5) = C(6, 1) = 6
     m = 3 : C(7, 5) = C(7, 2) = (7*6)/(2!) = 21
     m = 4 : C(8, 5) = C(8, 3) = (8*7*6)/(3!) = 56

Question

     Find next two numbers of this sequence from Pascal traingle

Answer

     Next two numbers are 126 and 252

    Go to Begin

    Q04 Number of terms in expansion of (a + b + c)^n

    Expansion of (a + b + c)^n has (n + 1)*(n + 2)/2
    • Expansion of (a + b + c)^0 has 1 term
    • Expansion of (a + b + c)^1 has 3 terms
    • Expansion of (a + b + c)^2 has 6 terms
    • Expansion of (a + b + c)^3 has 10 terms
    • Expansion of (a + b + c)^4 has 15 terms
    • Expansion of (a + b + c)^5 has 21 terms
    • Since the number of terms in triangular number pattern
    • They are in Pascal triangle for case r = 2
    • The mth term of triangular number pattern is m*(m + 1)/2
    • Hence the numer of terms of (a + b + n)^n is (n + 1)*(n + 2)/2
    • Verify for n = 5 : (5 + 1)*(5 + 2)/2 = 21
    Prove that expansion of (a + b + c)^3 has ten terms
    • The terms are
    • a^3, b^3, c^3
    • (a^2)*b, (a^2)*c, (b^2)*c, (b^2)*a, (c^2)*a, (c^2)*b
    • a*b*c
    • Hence it has 10 terms
    Reference
    • See polynomial coefficients in keyword P

    Go to Begin

    Q05 Number of terms in expansion of (a + b + c + d)^n

    Find number of terms in expansion of (a + b + c + d)^n
    • Expansion of (a + b + c + d)^0 has 1 term
    • Expansion of (a + b + c + d)^1 has 4 terms
    • Expansion of (a + b + c + d)^2 has 10 terms
    • Expansion of (a + b + c + d)^3 has 20 terms
    • Expansion of (a + b + c + d)^4 has 35 terms
    • Expansion of (a + b + c + d)^5 has 56 terms
    • Since the 1st difference of number of terms in triangular number pattern
    • They are in Pascal triangle for case r = 3
    • The mth term of triangular number pattern is m*(m + 1)*(m + 2)/6
    • Hence the numer of terms of (a + b + c + d)^n is (n + 1)*(n + 2)*(n + 3)/6
    Find expansion terms of (a + b + c + d)^2
    • The terms are
    • a^2, b^2, c^2, d^2
    • ab, ac, ad, bc, bd, cd
    • Hence it has 10 terms
    Verify that expansion terms of (a + b + c + d)^5 is 56
    • n = 5
    • Hence number of expansion terms = (5 + 1)*(5 + 2)*(5 + 3)/6 = 56
    • Home work : List the 56 terms
    Reference
    • Find coefficients of polynomial expansion
    • See polynomial coefficients in keyword P

    Go to Begin

    Q06 Number of terms in expansion of (a + b + c + d + e)^n

    Excercises : Use Pascal triangle (Hint : Use r = 4 case)
    • 1. Find number of terms in expansion of (a + b + c + d + e)^2
    • 2. Find number of terms in expansion of (a + b + c + d + e)^3
    • 3. Find number of terms in expansion of (a + b + c + d + e)^4
    • 4. Find number of terms in expansion of (a + b + c + d + e)^5

    Go to Begin
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