Question

The number of selection of $n$ objects from $2n$ objects of which $n$ are identical and the rest are different is

• A. $2^n$
• B. $2^{n - 1}$
• C. $2^{n} - 1$
• D. $2^{n - 1} + 1 $

Answer 1

Answer: (A)

Number of ways of selection of $n$ objects from $2 n$ objects, where as $n$ objects are identical in out of $2 n$ objects.
$n$ identical and no different object = 1 ways
$={ }^{n} C_{0}$
$n-1$ identical and 1 different object
$=1 \times{ }^{n} C _{1}$
$n-2$ identical and 2 different object
$=1 \times{ }^{n} C_{2}$
..............................
0 identical and $n$ different objects $=1 \times{ }^{n} C_{n}$
$={ }^{n} C_{0}+{ }^{n} C_{1}+{ }^{n} C_{2}+\ldots \ldots+{ }^{n} C_{n}=2^{n}$