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11 Jan 2022, 09:18
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
29% (02:29) correct
71%
(02:59)
wrong
based on 52
sessions
History
Date
Time
Result
Not Attempted Yet
The number of groups of three or more distinct numbers that can be chosen from 1, 2, 3, 4, 5, 6, 7 and 8 so that the groups always include 3 and 5, while 7 and 8 are never included together is:
A) 45
B) 47
C) 48
D) 60
E) 63
A) 45
B) 47
C) 48
D) 60
E) 63
28 Jan 2022, 07:35
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sujoykrdatta
We need to select 3 or more numbers from the numbers 1 to 8 such that 3 and 5 are definitely there.
The remaining numbers are: 1, 2, 4, 6, 7, 8.
We need to select 1 or more numbers from these.
We cannot select both 7 and 8.
Case 1: Let us select 7. We need to remove 8 and from the remaining 4 numbers, we can select none, one, two, three or all.
The number of ways = \(2^4\) = 16
Case 2: Let us select 8. We need to remove 7 and from the remaining 4 numbers, we can select none, one, two, three or all.
The number of ways = \(2^4\) = 16
Case 3: Let us remove 7 and 8. From the remaining 4 numbers, we can select one, two, three or all.
The number of ways = \(2^4 - 1\) = 15
Thus, total number of ways = 16 + 16 + 15 = 47
Answer B
Watch an Alternative approach in the YouTube video here:
28 Jan 2022, 07:45
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We don't care about the 3 and 5. If the question simply asked how many selections are possible from the remaining numbers, then for each of the numbers 1, 2, 4, 6, 7 and 8, we'd have two choices: include the number in our selection or exclude it. So we'd have 2^6 = 64 selections we could make in total where we pick any collection of these numbers. Now 1/4 of these selections include both the 7 and the 8 (and 1/4 include only the 7, 1/4 only the 8, and 1/4 neither), and we don't want to count those, leaving us with 48 selections. But we must select at least one number to get up to three numbers in total, so we also must exclude the one selection where we don't pick any of the numbers, and the answer is 47.
