Question

Let S={1,2,3,4,5,6}.

How many subsets of cardinality 4 contain at least one odd number?

Answer 1

15 subsets of cardinality 4 contain at least one odd number.

Explanation:

Here the given set,

S={1,2,3,4,5,6},

Since, a set having cardinality 4 having 4 elements,

The number of odd digits = 3 ( 1, 3, 5 )

And, the number of even digits = 3 ( 2, 4, 6 )

Thus, the total possible arrangement of a set having 4 elements out of which atleast one odd number =
`^3C_1* ^3C_3+^3C_2* ^3C_2+^3C_3* ^3C_1`

By using
`^nC_r=(n!)/(r!(n-r)!)`,

`=3* 1+3* 3+1* 3`

`=3+9+3`

`=15`

Hence, 15 subsets of cardinality 4 contain at least one odd number.