Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have to tell if the following inequality is true: $$\sum^{8564}_{i=82} \binom{8564}{i} < 2^{8564}$$

but how do I tackle that? I reckon the standard formula for calculate the value of the binomial coefficient is kind of useless in this case, the summation startles me. I feel really stuck, can anyone please throw any hint at me?

share|cite|improve this question
I believe the standard term is inequality instead of disequation. Am I the only one who has never heard of a disequation before? – Daan Michiels Feb 6 '13 at 16:27
You are absolutely right. I got the word wrong because of my native language. Editing right now! – haunted85 Feb 6 '13 at 16:33
up vote 4 down vote accepted

Observe that $$2^n=(1+1)^n=\sum_{k=0}^n\binom{n}{k}1^k1^{n-k}=\sum_{k=0}^n\binom{n}{k}$$

share|cite|improve this answer

Hint $$2^n = (1+1)^n = \sum_{i=0}^n \begin{pmatrix} n \\ i\end{pmatrix} 1^i 1^{n-i}$$

share|cite|improve this answer

Use the very well known, textbook identity $$\sum_0^n \tbinom{n}{k} = (1+1)^n = 2^n $$ Can you show that your LHS is strictly smaller than the LHS of the above equation.

share|cite|improve this answer

Hint: You don't need to calculate anything explicitly. What is the formula for $(1+x)^n$?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.