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:
![T o t a l space A r r a n g e m e n t s space equals fraction numerator X factorial over denominator left parenthesis x 1 factorial right parenthesis left parenthesis x 2 factorial right parenthesis left parenthesis x 3 factorial right parenthesis...... left parenthesis x n factorial right parenthesis end fraction](data:image/png;base64,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)
The correct answer is: ![fraction numerator 17 factorial over denominator 6 factorial 7 factorial 4 factorial end fraction](data:image/png;base64,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)
Detailed Solution
![I n space t h e space a b o v e space f o r m u l a comma space X space equals space t o t a l space n u m b e r space o f space e n t i t i e s comma space x subscript 1 space space a r e space t h e space s a m e space k i n d space o f space e n t i t i e s comma space t h a t space i s space t h e y space a r e space i d e n t i c a l comma space x subscript 2 space space a r e space a l s o space t h e space s a m e space k i n d space o f space e n t i t i e s space b u t space d i f f e r e n t space f r o m space x subscript 1 space. space
S i m i l a r l y comma space x subscript 3 comma x subscript 4 comma x subscript 5..... x subscript n space space a r e space t h e space n u m b e r space o f space i d e n t i c a l space e n t i t i e s space b u t space d i f f e r e n t space f r o m space e a c h space o t h e r.](data:image/png;base64,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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:
![T o t a l space A r r a n g e m e n t s space space equals space fraction numerator X factorial over denominator left parenthesis x subscript 1 factorial right parenthesis left parenthesis x subscript 2 factorial right parenthesis left parenthesis x subscript 3 factorial right parenthesis...... left parenthesis x subscript n factorial right parenthesis end fraction
I n space t h i s space f o r m u l a comma space X space i s space t h e space t o t a l space n u m b e r space o f space e n t i t i e s comma space x subscript 1 space space i s space t h e space n u m b e r space o f space i d e n t i c a l space e n t i t i e s space o f space t h e space f i r s t space k i n d comma space x subscript 2 space space i s space t h e space n u m b e r space o f space i d e n t i c a l space e n t i t i e s space o f space t h e space s e c o n d space k i n d comma space a n d space s o space o n. space
I n space o u r space c a s e comma space X equals 17 space comma x subscript 1 equals 7 space comma x subscript 2 equals 6 space space a n d space x subscript 3 equals 4 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)
Thus, we get,
![T o t a l space A r r a n g e m e n t s space equals space fraction numerator 17 factorial over denominator left parenthesis 7 factorial right parenthesis left parenthesis 6 factorial right parenthesis left parenthesis 4 factorial right parenthesis end fraction space equals space bold 17 bold cross times bold 14 bold cross times bold 13 bold cross times bold 11 bold cross times bold 10 bold cross times bold 3 bold cross times bold 4 bold space bold space bold equals bold 4084080 bold space bold space](data:image/png;base64,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)
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.
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.
Thus we can say that the permutation concerns both the selection and the arrangement of the selected things in all possible ways.