In an election the number of candidates is one more than the number of members to be elected. If a
voter can vote in 254 different ways, find the number of candidates.
(a) 8
(b) 10
(c) 7
(d) None of these
Answers
Answer:
(a) 8
Step-by-step explanation:
ANSWER
Let there be n candidates. Then
nC1+nC2+....+nCn−1=254
⇒2n−2=254
or 2n=28 or n=8
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Given : In an election the number of candidates is one more than the number of members to be elected.
a voter can vote in 254 different ways,
To Find :
number of candidates.
(a) 8
(b) 10
(c) 7
(d) None of these
Step-by-step explanation:
Candidate = N
Member to be selected = N
Voter can Choose 1 , 2 , 3 -----------------------or N-1 candidate
as N-1 members has to be elected ( can not be zero as he is voting)
1 candidate can be selected in ⁿC₁
2 candidate can be selected in ⁿC₂
3 candidate can be selected in ⁿC₃
n candidate can be selected in ⁿCₙ₋₁
Total ways = ⁿC₁ + ⁿC₂ + ⁿC₃ + ........................+ ⁿCₙ₋₁
Add and subtract ⁿC₀ + ⁿCₙ
ⁿC₁ + ⁿC₂ + ⁿC₃ + ........................+ ⁿCₙ₋₁ + ⁿC₀ + ⁿCₙ - ⁿC₀ - ⁿCₙ
ⁿC₀ = 1 , ⁿCₙ = 1
= ⁿC₀ + ⁿC₁ + ⁿC₂ + ⁿC₃ + ........................+ ⁿCₙ₋₁ + ⁿCₙ -1 - 1
= 2ⁿ - 2
2ⁿ - 2 = 254
=> 2ⁿ = 256
=> n = 8
number of candidates. = 8
option a is correct
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