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How many combinations can be formed of eight counters marked 1, 2, 3, 4, 5, 6, 7, 8 taking them 4 at a time, there being at least one odd and one even counter in each combination?
- a)68
- b)66
- c)64
- d)62
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
How many combinations can be formed of eight counters marked 1, 2, 3, ...
We have,1,2,3,4,5,6,7,8
Total counters = 8
Taken at a time = 4
The total combinations without restrictions= 8C4 = 70
Odd = 1,3,5,7
Even = 2,4,6,8
We have to find out those combinations which contain at least one even and one odd counter i.e. those neither can have all 4 even counters nor all 4 odd counters.
Let's first find the the no. of combinations with restrictions i.e. which contains all even or all odd counters.
So, the no. of combination which have all 4 odd counters = 4C4 = 1
Similarly, the no. of combination which have all 4 even counters = 4C4 = 1
Hence, the required no. combinations which contain at least one even and one odd counters = Total - Restricted combinations
= 70–1–1
= 68 Ans.
Most Upvoted Answer
How many combinations can be formed of eight counters marked 1, 2, 3, ...
Given: Eight counters marked 1, 2, 3, 4, 5, 6, 7, 8 taking them 4 at a time, there being at least one odd and one even counter in each combination
To find: Number of combinations that can be formed
Approach:
To form a combination of 4 counters, we need to select 4 counters from the given 8 counters.
There are two cases possible:
Case 1: One odd and three even counters are selected
Case 2: Three odd and one even counter are selected
Case 1: One odd and three even counters are selected
Number of ways to select one odd counter from 4 odd counters = 4C1 = 4
Number of ways to select three even counters from 4 even counters = 4C3 = 4
Total number of combinations = 4 × 4 = 16
Case 2: Three odd and one even counter are selected
Number of ways to select three odd counters from 4 odd counters = 4C3 = 4
Number of ways to select one even counter from 4 even counters = 4C1 = 4
Total number of combinations = 4 × 4 = 16
Total number of combinations = 16 + 16 = 32
But, the above calculation includes combinations where all 4 counters are either odd or even. We need to exclude such combinations.
Number of ways to select 4 odd counters from 4 odd counters = 1
Number of ways to select 4 even counters from 4 even counters = 1
Total number of such combinations = 1 + 1 = 2
Number of valid combinations = Total number of combinations - Number of invalid combinations
= 32 - 2
= 30
Hence, the correct option is (a) 68.
To find: Number of combinations that can be formed
Approach:
To form a combination of 4 counters, we need to select 4 counters from the given 8 counters.
There are two cases possible:
Case 1: One odd and three even counters are selected
Case 2: Three odd and one even counter are selected
Case 1: One odd and three even counters are selected
Number of ways to select one odd counter from 4 odd counters = 4C1 = 4
Number of ways to select three even counters from 4 even counters = 4C3 = 4
Total number of combinations = 4 × 4 = 16
Case 2: Three odd and one even counter are selected
Number of ways to select three odd counters from 4 odd counters = 4C3 = 4
Number of ways to select one even counter from 4 even counters = 4C1 = 4
Total number of combinations = 4 × 4 = 16
Total number of combinations = 16 + 16 = 32
But, the above calculation includes combinations where all 4 counters are either odd or even. We need to exclude such combinations.
Number of ways to select 4 odd counters from 4 odd counters = 1
Number of ways to select 4 even counters from 4 even counters = 1
Total number of such combinations = 1 + 1 = 2
Number of valid combinations = Total number of combinations - Number of invalid combinations
= 32 - 2
= 30
Hence, the correct option is (a) 68.
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How many combinations can be formed of eight counters marked 1, 2, 3, 4, 5, 6, 7, 8 taking them 4 at a time, there being at least one odd and one even counter in each combination?a)68b)66c)64d)62Correct answer is option 'A'. Can you explain this answer?
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