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Question

$A$ set $B$ contain 2007 elements. Let $C$ be the set con sisting of subsets of $B$ which contain at most $1003$
elements. The number of elements in $C$ is

Moderate

Solution

The number of required subsets is
$\begin{array}{l} N =^{2007} C_{0} +^{2007} C_{1} +^{2007} C_{2} + \dots +^{2007} C_{1003} \\ N =^{2007} C_{2007} +^{2007} C_{2006} + \dots +^{2007} C_{1004} \end{array}$
Adding above two equations we get
$\begin{array}{l} 2 N =^{2007} C_{0} +^{2007} C_{1} + \dots +^{2007} C_{2007} \\ \Rightarrow N = 2^{2006} \end{array}$

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Combinations

Remember concepts with our Masterclasses.

A set B contain 2007 elements. Let C be the set con sisting of subsets of B which contain at most 1003 elements. The number of elements in C is

The number of required subsets is N=2007C0+2007C1+2007C2+…+2007C1003N=2007C2007+2007C2006+…+2007C1004Adding above two equations we get 2N=2007C0+2007C1+…+2007C2007⇒N=22006

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$A$ set $B$ contain 2007 elements. Let $C$ be the set con sisting of subsets of $B$ which contain at most $1003$
elements. The number of elements in $C$ is