Properties and Applications of Binomial Coefficients: Quick Revision

A brief note on Properties and Applications of Binomial Coefficients. Read it before a test. Try each question at the end before you open its answer.

The rules

• If ${}^{n}C_{a} = {}^{n}C_{b}$ with $a \ne b$, then $a + b = n$.
• For a polynomial $f$: put $x = 1$ for the total and $x = -1$ for the alternating sum.
• $\sum r \cdot {}^{n}C_{r} = n \cdot 2^{n - 1}$.
• $(d + 1)^n$ leaves remainder $1$ on division by $d$, and $(d - 1)^n$ leaves $1$ or $d - 1$.
• Modulo $d^2$, $(d + 1)^n \equiv 1 + nd$.
• For small $h$, $(1 + h)^n \approx 1 + nh + {}^{n}C_{2}h^2$.

A worked example

Find the sum of all the coefficients, and the sums of the even and the odd powers of $x$, in the expansion of $(2x + 5)^4$.

Step 1. The total. Put $x = 1$: $7^4 = 2401$.

Step 2. The alternating sum. Put $x = -1$: $3^4 = 81$.

Step 3. The two halves. The even powers add up to $\dfrac{2401 + 81}{2} = 1241$, and the odd powers to $\dfrac{2401 - 81}{2} = 1160$.

Answer: $2401$, $1241$ and $1160$.

Practice with new numbers

A program to try

The program finds the sums of the worked example with $x = 1$ and $x = -1$, and checks them against the expanded coefficients.

Mistakes to avoid

• Expanding first. To add coefficients, put $x = 1$.
• A half-sum of $2^n$. Each half is $2^{n - 1}$.
• A negative remainder. Add the divisor to make it positive.
• A large $h$. A short sum works only when $h$ is small.
• Upside-down ratio. The next coefficient is the one before times $\dfrac{n - r + 1}{r}$.

Check yourself

1. Find ${}^{16}C_{14}$.

2. Find the sum of all the coefficients in $(5x - 2)^{3}$.

3. Find the remainder when $8^{30}$ is divided by $7$.

4. Estimate $1.03^{6}$ with two terms.

5. If ${}^{n}C_{3} = {}^{n}C_{9}$, find $n$.