⦿ The total number of combination of n different things taken one or more at a time = 2ⁿ – 1
Proof:-
Method (1):-
Total distinct things = n
for ‘1ˢᵗ’ thing there are 2 ways of selection: included or excluded
similarly, for ‘2ⁿᵈ’ thing there are 2 ways of selection: included or excluded
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Similarly, for ‘nᵗʰ’ thing there are 2 wayss of selection: included or excluded
Hence total number of ways of selection:
= 2×2×2×………….n times
= 2ⁿ
Here point to remember is that in these 2ⁿ ways, there is also 1 way in which none of the article will be included & we are to select one or more article
Hence number of ways = 2ⁿ – 1
Method (2):-
number of ways of selecting 1 thing out of n things = ⁿC₁
number of ways of selecting 2 thing out of n things = ⁿC₂
number of ways of selecting 3 thing out of n things = ⁿC₃
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number of ways of selecting n thing out of n things = ⁿCₙ
∴ total number of ways = ⁿC₁ + ⁿC₂ + ⁿC₃ + ………. + ⁿCₙ
& by property of binomial coefficient we know that value of aboveww expression is 2ⁿ – 1
Hence number of ways = 2ⁿ – 1
Example:- Amit, a teacher in a school, has 15 students in his class. He has to select 1 or more students for the laboratory work. then in how many ways he can do this?
Solution:- Here n = 15
⇒ 2¹⁵ – 1
= 32768 – 1
= 32767 ways Answer