Question
The number of ways in which 17 billiard balls be arranged in a row if 7 of them are black, 6 are red, 4 are white is
Hint:
To solve this question, we will assume that all the balls of the same color are identical. Now, to arrange all these balls, we will apply the following formula:
The correct answer is:
Detailed Solution
Before solving the question, we are going to assume that all the balls of the same color are identical. This means that one red ball is identical to another red ball. Similarly, one black ball is identical to all the other black balls and one white ball will be similar to other white balls. Now, we are given that out of 17 balls, 7 of them are black, 6 are red and 4 are white. Now, we will arrange these balls in a row. The formula by which we can arrange the total number of entities which contain similar entities is given as:
Thus, we get,
Thus, there are 4084080 ways in which we can arrange these billiard balls.
We cannot arrange the billiards balls as follows:
There are 17 balls, so the total number of arrangements = 17!
This is incorrect because this method is applicable only when the balls are distinct, not identical. But, in our case, all the balls of the same color are identical.
Related Questions to study
Number of permutations that can be made using all the letters of the word MATRIX is
Number of permutations that can be made using all the letters of the word MATRIX is
The number of different signals that can be made by 5 flags from 8 flags of different colours is :
Each of the different arrangements which can be made by taking some or all of a number of things at a time is called permutation. The number of permutation without any repetition states that arranging n objects taken r at a time is equivalent to filling r places out of n things =n(n−1)(n−2)......(n−r−1) ways = 
Thus we can say that the permutation concerns both the selection and the arrangement of the selected things in all possible ways.
The number of different signals that can be made by 5 flags from 8 flags of different colours is :
Each of the different arrangements which can be made by taking some or all of a number of things at a time is called permutation. The number of permutation without any repetition states that arranging n objects taken r at a time is equivalent to filling r places out of n things =n(n−1)(n−2)......(n−r−1) ways =
Thus we can say that the permutation concerns both the selection and the arrangement of the selected things in all possible ways.
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