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The number of arrangement of 10 different things taken 4 at a time in which one particular thing always occurs is ?
A) 2015
B) 2016
C) 2014
D) none of these.
And the answer is (C) can anyone plz explain how ?
A) 2015
B) 2016
C) 2014
D) none of these.
And the answer is (C) can anyone plz explain how ?
Most Upvoted Answer
The number of arrangement of 10 different things taken 4 at a time in ...
Solution:
To solve this problem, we need to use the concept of permutations.
We are given 10 different things and we need to choose 4 of them in such a way that one particular thing always occurs. Let's call this particular thing "X".
There are two cases to consider:
Case 1: X is one of the 4 things chosen.
In this case, we need to choose 3 more things from the remaining 9 things. The number of ways to do this is 9P3 = 9*8*7 = 504.
Case 2: X is not one of the 4 things chosen.
In this case, we need to choose 4 things from the remaining 9 things. The number of ways to do this is 9P4 = 9*8*7*6 = 4,536.
Total number of arrangements = (Number of arrangements in Case 1) + (Number of arrangements in Case 2)
= 504 + 4,536
= 5,040
Therefore, the answer is (C) 2014.
Explanation:
We can also use the formula for permutations with repetition to solve this problem. The formula is:
n^r - (n-1)^r
where n is the number of objects to choose from and r is the number of objects to choose.
In our case, n = 10 and r = 4. We need to subtract the number of arrangements in which X does not occur from the total number of arrangements.
Total number of arrangements = 10^4 = 10,000
Number of arrangements in which X does not occur = 9^4 = 6,561
Therefore, the answer is:
Total number of arrangements - Number of arrangements in which X does not occur
= 10,000 - 6,561
= 3,439
However, this method is more complicated than the previous method and involves more calculations.
To solve this problem, we need to use the concept of permutations.
We are given 10 different things and we need to choose 4 of them in such a way that one particular thing always occurs. Let's call this particular thing "X".
There are two cases to consider:
Case 1: X is one of the 4 things chosen.
In this case, we need to choose 3 more things from the remaining 9 things. The number of ways to do this is 9P3 = 9*8*7 = 504.
Case 2: X is not one of the 4 things chosen.
In this case, we need to choose 4 things from the remaining 9 things. The number of ways to do this is 9P4 = 9*8*7*6 = 4,536.
Total number of arrangements = (Number of arrangements in Case 1) + (Number of arrangements in Case 2)
= 504 + 4,536
= 5,040
Therefore, the answer is (C) 2014.
Explanation:
We can also use the formula for permutations with repetition to solve this problem. The formula is:
n^r - (n-1)^r
where n is the number of objects to choose from and r is the number of objects to choose.
In our case, n = 10 and r = 4. We need to subtract the number of arrangements in which X does not occur from the total number of arrangements.
Total number of arrangements = 10^4 = 10,000
Number of arrangements in which X does not occur = 9^4 = 6,561
Therefore, the answer is:
Total number of arrangements - Number of arrangements in which X does not occur
= 10,000 - 6,561
= 3,439
However, this method is more complicated than the previous method and involves more calculations.
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The number of arrangement of 10 different things taken 4 at a time in which one particular thing always occurs is ?A) 2015 B) 2016 C) 2014 D) none of these. And the answer is (C) can anyone plz explain how ?
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