5
Question
A set contains (2n+1) elements. The number of sub-sets of the set which
contain at most n elements is
A set contains (2n+1) elements. The number of sub-sets of the set which
contain at most n elements is
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Solution
The correct option is D
The number of sub-sets of the set which contain at most n elements is
2n+1C0+2n+1C1+.........+2n+1Cn = S (Say)
Then 2S = 2(2n+1C0+2n+1C1+.........+2n+1Cn)
= (2n+1C0+2n+1C2n+1)+(2n+1C1+2n+1C2n)+......+(2n+1Cn+2n+1Cn+1) {∵nCr=nCn−r}
= 2n+1C0+2n+1C1+..........+2n+1C2n+1 =22n+1
⇒S = 22n
The number of sub-sets of the set which contain at most n elements is
2n+1C0+2n+1C1+.........+2n+1Cn = S (Say)
Then 2S = 2(2n+1C0+2n+1C1+.........+2n+1Cn)
= (2n+1C0+2n+1C2n+1)+(2n+1C1+2n+1C2n)+......+(2n+1Cn+2n+1Cn+1) {∵nCr=nCn−r}
= 2n+1C0+2n+1C1+..........+2n+1C2n+1 =22n+1
⇒S = 22n