Calculate or approximate $\sum\limits_{k=801}^{849}{ \binom {2400} {k}} $

Question

Is there any formula which can help me to calculate directly the following sum :

$$\sum_{k=801}^{849} \binom {2400} {k} \text{ ? } $$

Or can you help me for an approximation?

Answer 1

$$ \sum_{k=801}^{\color{red}{849}}{ \binom {\color{blue}{2400}} {k}} \approx\binom {\color{blue}{2400}} {\color{red}{849}}\cdot\frac1{1-\frac{\color{red}{849}}{\color{blue}{2400}-\color{red}{849}}}=\binom {2400} {849}\cdot\frac{1551}{702} $$ The RHS is approximately $6.323\cdot10^{675}$. The LHS is approximately $6.302\cdot10^{675}$. The relative error is less than $0.5\%$.

Answer 2

This can be computed using Sum([801..849],k->Binomial(2400,k)); in GAP, which gives:

6301658589252789789015394548565753645266501707773027193237154277135560724763449264321985233013302083482867404033075780563717296980953719800345900949177485867102506636266227692550924263009975743334850017707243878849834456104549363646796461925922393903588452037583051589363088432111720742572219650413596433425388801991237454447814199442671137445437352797716466882851664850702263146918207481280464655954611009667731242187491732271534642120317745100700127873871822642415935966102219060535417049206248240674733546894430281964131727432744187261341581666985170681694633765243107624160397671919189350311078072089739085127819403410647241830980073127572833414340863527184811939282852420

This is approximately $6.30 \times 10^{675}$.