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  4. The number of one element subset, the number of two element subsets and the number of three element subsets of a set containing more than three elements are consecutive terms of an A.P. The number of elements in the set can be,

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Q. The number of one element subset, the number of two element subsets and the number of three element subsets of a set containing more than three elements are consecutive terms of an A.P. The number of elements in the set can be,

Permutations and Combinations

Solution:

Given ${ }^{ n } C _1,{ }^{ n } C _2,{ }^{ n } C _3 \longrightarrow$ A.P.
$ 2 \cdot{ }^{ n } C_2={ }^{ n } C_1+{ }^{ n } C_3 $
$ \frac{2 n ( n -1)}{2}= n +\frac{ n ( n -1)( n -2)}{6} $
$\Rightarrow n ^2-9 n +14=0 $
$\therefore n =7 $