Useful Binomial Identities

The Binomial Theorem gives several important identities.

Sum of Binomial Coefficients

Set $a=1$ and $b=1$ in the Binomial Theorem:

Therefore,

This says that the sum of the entries in row $n$ of Pascal's Triangle is $2^n$.

Alternating Sum

Set $a=1$ and $b=-1$

For $n>0$, this gives

This means the sum of the even indexed binomial coefficients equals the sum of the odd indexed binomial coefficients.

Weighted Sum

Another useful identity is

One way to see this is combinatorially, count the number of ways to choose a committee from $n$ people and then choose a leader from that committee.

If the committee has size $k$, there are

ways.

On the other hand, choose the leader first in $n$ ways, and then choose any subset of the remaining $n-1$ people to join the committee. This gives

ways.

So

Estimating and Approximating Powers

The Binomial Theorem can also be used to approximate powers.

For small values of $x$,

If $x$ is very small, the higher powers of $x$ may be much smaller than the first few terms.

For example,

The first few terms already give a strong approximation

In olympiad problems, this idea is more commonly used for exact expansion than decimal approximation, but the intuition is still useful.

Modular Applications

The Binomial Theorem is often useful in modular arithmetic.

Example 5

Find the remainder when $101^5$ is divided by $100$.

Write

Then

Using the Binomial Theorem,

Every term except the first is divisible by $100$, so

Thus the remainder is $1$.

Example 6

Show that $10^{n}-1$ is divisible by $9$ for every positive integer $n$.

Write

Then

By the Binomial Theorem,

Subtracting $1$ gives

Every term on the right is divisible by $9$, so $10^n-1$ is divisible by $9$.

Combinatorial Interpretation

The Binomial Theorem is not just an algebraic formula. It also has a counting interpretation.

When expanding

we are multiplying $n$ copies of

Each factor gives us two choices: choose $a$ or choose $b$.

Thus the full expansion corresponds to all possible ways of making $n$ binary choices.

The coefficient

counts the number of ways to choose exactly $k$ of the factors to contribute $b$.

This is why binomial coefficients appear naturally in combinatorics and probability.

Application to Counting

Suppose there are $n$ people, and each person can either be chosen for a team or not chosen.

There are $2$ choices for each person, so the total number of possible teams is

On the other hand, we can count by team size.

There are

teams of size $0$,

teams of size $1$,

teams of size $2$,

and so on.

Therefore,

This is the identity obtained from expanding $(1+1)^n$.

Application to Probability

The Binomial Theorem is also the foundation of binomial probability.

Suppose an experiment has two possible outcomes

• Success with probability $p$
• Failure with probability $1-p$

If the experiment is repeated $n$ times independently, the probability of getting exactly $k$ successes is

Why?

We must choose which $k$ of the $n$ trials are successes. This can be done in

ways.

For any fixed set of $k$ successes and $n-k$ failures, the probability is

Therefore the total probability is

If we sum over all possible values of $k$, we get

By the Binomial Theorem this equals

This confirms that the probabilities of all possible numbers of successes sum to $1$.

Worked Probability Example

Example 7

A fair coin is flipped $6$ times. What is the probability of getting exactly $4$ heads?

Here, a success is getting heads, so

We want exactly $4$ successes in $6$ trials, so

The probability is

Since

this becomes

So the probability is

Key Contest Insight

The Binomial Theorem is useful whenever a problem involves a power of a sum. Look for expressions such as

or

The theorem can help you:

• Expand powers efficiently
• Find a single coefficient or term
• Prove divisibility statements
• Evaluate binomial sums
• Count subsets
• Compute probabilities

The main idea is that coefficients in expansions count choices.

Strategy Tips

• Identify the two terms being raised to a power
• Write the general term before expanding everything
• For coefficient problems, solve for the exponent you need
• For constant term problems, set the exponent of the variable equal to $0$
• For modular arithmetic, write numbers as something convenient plus a small remainder, such as $101=100+1$
• For combinatorics, remember that $\binom{n}{k}$ counts choices
• For probability, identify the number of successes, the success probability, and the number of trials

Common Pitfalls

• Forgetting the binomial coefficient
• Mixing up $k$ and $n-k$ in the general term
• Losing negative signs in expansions such as $(a-b)^n$
• Expanding the whole expression when only one term is needed
• Forgetting to check whether a term is constant in constant-term problems
• In probability, forgetting to multiply by the number of ways to choose which trials are successes

Practice Problems