Subject
Quantitative Aptitude
Topic
Permutation & Combination
Six different pencils and three bags are taken. Each pencil is to be put in one of these bags. No bag should remain empty and all bags should not have same number of pencils. In how many ways can this be done?
Six different pencils and three bags are taken. Each pencil is to be put in one of these bags. No bag should remain empty and all bags should not have same number of pencils. In how many ways can this be done?
ক)
729খ)
537গ)
444ঘ)
637ঙ)
669Explanation
Each pencil can be put in one of three bags. ⇒ Number of ways in which this can be done = 3 × 3 × 3 × 3 × 3 × 3 = 729 Out of these, there can be some cases when one or more bags are empty, and when all bags have same number of pencils. Case i) one bag is empty Empty bag can be selected in 3 c 1 ways i.e. 3 ways Now, each pencil can be put in one of two ways. Number of ways in which this can be done = 3 × (2 × 2 × 2 × 2 × 2 × 2) = 192 Case ii) two bags are empty All pencils will be in one of three bags. This can be done in only 3 ways. Case iii) three bags are empty. This case is not possible Case iv) all bags have same number of pencils, i.e., 2 pencils. ⇒ We have to select two pencils for each bag. Number of ways in which this can be done = 6 c 2 × 4 c 2 × 2 c 2 = 15 × 6 × 1 = 90 ∴ Total number of ways in which this can be done = 729 – 192 – 3 – 90 = 444Related questions
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