25

THE BINOMIAL THEOREM

Formal statement of the theorem

Pascal's triangle

THE BINOMIAL THEOREM shows how to calculate a power of a binomial—(a + b)ⁿ—without actually multiplying.

For example, if we actually multiplied out the 4th power of (a + b) --

(a + b)⁴ = (a + b)(a + b)(a + b)(a + b)

-- then on collecting like terms we would find:

(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ .  .  .  .  (1)

Note:  The literal factors are all possible terms in a and b where the sum of the exponents is 4:  a⁴,  a³b,  a²b²,  ab³,  b⁴.

The degree of each term is 4.

The first term is actually a⁴b⁰, which is a⁴ · 1.

Thus to "expand" (a + b)⁵, we would anticipate the following terms, in which the sum of all the exponents is 5:

(a + b)⁵ =  ? a⁵ +  ? a⁴b +  ? a³b² +  ? a²b³ +  ? ab⁴ +  ? b⁵

The question is, What are the coefficients?

They are called the binomial coefficients.  In the expansion of
(a + b)⁴, the binomial coefficients are

1  4  6  4  1

line (1) above.

  Note the symmetry:  The coefficients from left to right are the same right to left.

The answer to the question, "What are the binomial coefficients?" is called the binomial theorem.  It shows how to calculate the coefficients in the expansion of (a + b)ⁿ.

The symbol for a binomial coefficient is .  The upper index n is the exponent of the expansion; the lower index k indicates which term, starting with k = 0.

For example, when n = 5, each term in the expansion of  (a + b)⁵  will look like this:

a^{5 − k}b^k

k will successively take on the values 0 through 5.

(a + b)⁵ = a⁵  +  a⁴b  +  a³b²  +  a²b³  +   ab⁴  +  b⁵

Note:  Each lower index is the exponent of b.  The first term has k = 0 because in the first term, b appears as b⁰, which is 1.

Now, what are these binomial coefficients, ?

The theorem states that the binomial coefficients are none other than the combinatorial numbers, ⁿC_k .

(a + b)5	=	5C0a5 + 5C1a4b + 5C2a3b2 + 5C3a2b3 + 5C4ab4 + 5C5b5
	=	1a5 + a4b + a3b2 + a2b3 + ab4 + b5
	=	a5  +  5a4b  +  10a3b2  +  10a2b3  +  5ab4  +  b5

The binomial coefficients here are

1  5  10  10  5  1.

Note the symmetry.

The coefficient of the first term is always 1, and the coefficient of the second term is the same as the exponent of (a + b), which here is 5.

Using sigma notation and factorials for the combinatorial numbers, here is the binomial theorem:

What follows the summation sign is the general term.  Each term in the sum will look like that—the first term having k = 0; then k = 1, k = 2, and so on, up to k = n.

Notice that the sum of the exponents  (n − k) + k, always equals n.

Example 1.

a)  The term  a⁸b⁴  occurs in the expansion of what binomial?

Answer.   (a + b)¹².  The sum of 8 + 4 is 12.

b)  In that expansion, what number is the coefficient of a⁸b⁴?

 Answer.   It is the combinatorial number,

Note again: The lower index, in this case 4, is the exponent of b.

This same number is also the coefficient of  a⁴b⁸,  since  ¹²C₈ = ¹²C₄.

Example 2.   Expand (a − b)⁵.

 Solution.   We found the binomial coefficients to be  1  5  10  10  5  1.  The difference with (a − b) is that the signs of the terms will alternate:

(a − b)⁵ = a⁵ − 5a⁴b + 10a³b² − 10a²b³ + 5ab⁴ − b⁵.

For,  a − b = a + (−b),  therefore each term will have the form

a^{5 − k}(−b)^k.

When k is even, (−b)^k will be positive.  But when k is odd, (−b)^k will be negative.  

Each odd power of b will have a minus sign.

Example 3.   In the expansion of  (x − y)¹⁵, calculate the coefficients of x³y¹² and  x²y¹³.

 Solution.   The coefficient of  x³y¹²  is positive because the exponent of

y is even.  That coefficient is ¹⁵C₁₂.  But ¹⁵C₁₂ = ¹⁵C₃, and so we have

The coefficient of  x²y¹³, on the other hand, is negative because the exponent of y is odd.  The coefficient is  − ¹⁵C₁₃  =  − ¹⁵C₂.  We have

Example 4.   Write the first four terms of (a + b)ⁿ.  Do not use factorials.

Solution.

(a + b)n	=	nC0an  +  nC1an − 1b  +  nC2an − 2b2  +  nC3an − 3b3
	=	an + nan − 1b + an − 2b2 + an − 3b3.

Notice:  Each coefficient is a factor of the next coefficient.  The coefficient n is a factor of

.

That in turn is a factor of

.

To construct the next coefficient, then, multiply the present coefficient by the exponent of a in that term—

an − 3b3

-- namely n − 3:

And divide it by 1 more than the exponent of b.

That is the coefficient of a^{n − 4}b⁴.

Example 5.   Use the binomial theorem to expand  (a + b)⁸.

 Solution.   The expansion will begin :

(a + b)⁸ = a⁸ + 8a⁷b

The first coefficient is always 1.  The second is always the exponent of the expansion, in this case 8.

The next coefficient can be constructed as described above.  It will be the present coefficient, 8 --

(a + b)⁸ = a⁸ + 8a⁷b + 28a⁶b²

-- times the exponent of a in that term , 7, divided by 1 more then the exponent of b.  It will be 8· 7/2 = 28.

The next coefficient --

(a + b)⁸ = a⁸ + 8a⁷b + 28a⁶b² + 56a⁵b³

-- is 28· 6, divided by 3:

28· 6 / 3 = 28· 2 = 56.

The next --

(a + b)⁸ = a⁸ + 8a⁷b + 28a⁶b² + 56a⁵b³ + 70a⁴b⁴

-- is 56· 5, divided by 4:

56· 5 / 4 = 14· 5 = 70.

We have now come to the point of symmetry.  For, the coefficient of a³b⁵ is equal to the coefficient of a⁵b³, which is 56.  And so on for the remaining coefficients.

Here is the complete expansion:

Problem 1.   In the expansion of (a + b)ⁿ, each term has the form

a^{n − k}b^k,

where k successively takes on the value 0, 1, 2, . . ., n.

is the symbol for the binomial coefficient.

The binomial theorem is the statement that   = ?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

The combinatorial number, ⁿC_k.

Problem 2.   Use factorials to write the general term in the expansion of (a + b)ⁿ.

Problem 3.   

a)   Calculate the coefficient of  a⁴b⁶  in the expansion of (a + b)¹⁰.

¹⁰C₆ = ¹⁰C₄ = 210

b)   The coefficient of which other term is the same?   a⁶b⁴

c)   In the expansion of  (a + b)ⁿ, the coefficient of  a^{n − k}b^k  is the same
c)   as the coefficient of which other term?   a^kb^{n − k}

Problem 4.   Calculate the coefficient of

a)   x¹⁷y³ in the expansion of (x + y)²⁰.   1140

b)   x³y¹⁷ in the expansion of (x + y)²⁰.   1140

c)   x³y¹⁷ in the expansion of (x − y)²⁰.   −1140

d)   x²y¹⁸ in the expansion of (x − y)²⁰.   190

e)   x⁵y⁵ in the expansion of (x − y)¹⁰.   −252

f)   x¹⁰ in the expansion of (x − 1)¹⁵.   −3003

Problem 5.   Write the first four terms of (x + h)ⁿ.  Do not use factorials.

Problem 6.   Compute the first four terms of each of the following.

a)   (a + b)¹⁵   a¹⁵ + 15a¹⁴b + 105a¹³b² + 455a¹²b³

b)   (x − 1)²⁰   x²⁰ − 20x¹⁹ + 190x¹⁸ − 1140x¹⁷

Example 6.   Write the 5th term in the expansion of (a + b)¹⁰.

 Solution.   In the 1st term, k = 0.  In the 2nd term, k = 1.  And so on.

The index k—the exponent of b—of each term
is one less than the ordinal number of the term.

Thus in the 5th term, k = 4.  The exponent of b is 4.  The 5th term is

Problem 7.   Consider the expansion of (x + b)³⁰.

a)   What is the exponent of b in the 25th term?   24 

b)   In the kth term?   k − 1

c)   Write the fourth term, with its coefficient.   4,060x²⁷b³

Problem 8.   Calculate each of the following.

a)   The third term of (a + b)¹¹.   55a⁹b²

b)   The fifth term of (x − y)⁷.   35x³y⁴

c)   The tenth term of (x − 1)¹².   −220x³

Pascal's triangle

That triangular array is called Pascal's Triangle.  Each row gives the combinatorial numbers, which are the binomial coefficients.  That is, the row  1  2  1  are the combinatorial numbers ²C_k, which are the coefficients of (a + b)².  The next row—

1  3  3  1

—are the coefficients of (a + b)³; and so on.  

To construct the triangle, write 1, and below it write 1  1.  Begin and end each successive row with 1.  To construct the intervening numbers, add the two numbers immediately above.

Thus to construct the third row, begin it with 1, then add the two numbers immediately above:  1 + 1.  Write 2.  Finish the row with 1.  

To construct the next row, begin it with 1, and add the two numbers immediately above: 1 + 2.  Write 3.  Again, add the two numbers immediately above:  2 + 1 = 3.  Finish the row with 1.

Example 7.   Expand  (x − 1)⁶.

 Solution.   According to Pascal's triangle, the coefficients are

1  6  15  20  15  6  1.

In the binomial, x is "a", and −1 is "b".  The signs will alternate:

Example 8.   Expand  (x + 2)³.

 Solution.   The coefficients are  1  3  3  1.   x is "a", and 2 is "b".

Problem 9.   Use Pascal's triangle to expand the following.

a)   (a + b)³  =  a³ + 3a²b + 3ab² + b³

b)   (a − b)³  =  a³ − 3a²b + 3ab² − b³

c)   (x + y)⁴  =  x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

d)   (x − y)⁴  =  x⁴ − 4x³y + 6x²y² − 4xy³ + y⁴

e)   (x − 1)⁵  =  x⁵ − 5x⁴ + 10x³ − 10x² + 5x − 1

f)   (x + 2)⁵  =  x⁵ + 10x⁴ + 40x³ + 80x² + 80x + 32

g)   (2x − 1)³  =  8x³ − 12x² + 6x − 1

h)   (1 − xy)⁷ 

=  1 − 7xy + 21x²y² − 35x³y³ + 35x⁴y⁴ − 21x⁵y⁵ + 7x⁶y⁶ − x⁷y⁷

In the following Topic we will explain why the binomial coefficients are the combinatorial numbers. That will constitute a proof of the binomial theorem.

Next Topic:  Multiplication of sums

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