Question:
There are 15 identical balls for sale among which 4 are red, 5 are black and 6 are white. If a person buys at least one ball out of these 15 balls, then the total number of ways in which that person can buy the balls is
There are 15 identical balls for sale among which 4 are red, 5 are black and 6 are white. If a person buys at least one ball out of these 15 balls, then the total number of ways in which that person can buy the balls is
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When identical objects of different types are selected, count the possible quantities of each type independently and apply the multiplication principle. If at least one object must be chosen, subtract the case where all quantities are zero.
Updated On: Jul 29, 2026
- \(120\)
- \(119\)
- \(210\)
- \(209\)
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The Correct Option is D
Solution and Explanation
Concept:
Since balls of the same colour are identical, a selection is completely determined by the number of red, black and white balls chosen.
Apply the multiplication principle and then exclude the case of selecting no ball.
Step 1: Find the number of choices for each colour. Let \[ r=\text{number of red balls chosen}, \] \[ b=\text{number of black balls chosen}, \] \[ w=\text{number of white balls chosen}. \] Since there are \(4\) red balls, \[ r=0,1,2,3,4. \] Hence, the number of choices for \(r\) is \[ 5. \] Similarly, for \(5\) black balls, \[ b=0,1,2,3,4,5, \] giving \[ 6 \] choices. For \(6\) white balls, \[ w=0,1,2,3,4,5,6, \] giving \[ 7 \] choices.
Step 2: Find the total number of selections. By the multiplication principle, \[ 5\times 6\times 7=210. \] Thus, there are \(210\) possible selections including the selection of no ball.
Step 3: Exclude the case of selecting no ball. The selection \[ (r,b,w)=(0,0,0) \] corresponds to buying no ball. Since at least one ball must be purchased, this case is excluded. Therefore, \[ 210-1=209. \] \[ \boxed{\text{Total number of ways}=209} \] \[ \boxed{\text{Answer = (D)}} \]
Step 1: Find the number of choices for each colour. Let \[ r=\text{number of red balls chosen}, \] \[ b=\text{number of black balls chosen}, \] \[ w=\text{number of white balls chosen}. \] Since there are \(4\) red balls, \[ r=0,1,2,3,4. \] Hence, the number of choices for \(r\) is \[ 5. \] Similarly, for \(5\) black balls, \[ b=0,1,2,3,4,5, \] giving \[ 6 \] choices. For \(6\) white balls, \[ w=0,1,2,3,4,5,6, \] giving \[ 7 \] choices.
Step 2: Find the total number of selections. By the multiplication principle, \[ 5\times 6\times 7=210. \] Thus, there are \(210\) possible selections including the selection of no ball.
Step 3: Exclude the case of selecting no ball. The selection \[ (r,b,w)=(0,0,0) \] corresponds to buying no ball. Since at least one ball must be purchased, this case is excluded. Therefore, \[ 210-1=209. \] \[ \boxed{\text{Total number of ways}=209} \] \[ \boxed{\text{Answer = (D)}} \]
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