In how many ways can 7 distinct objects be divided among three people so that either one or two of them do not get any object?

• Case 1: Two guys don't get anything. All items go to 1 person. Hence, 3 ways.
  
  Case 2: 1 guy doesn't get anything. Division of 7 items among 2 guys. Lets say they are called A & B.
  
  a) A gets 6. B gets 1. 7 ways.
  b) A gets 5. B gets 2. 21 ways.
  c) A gets 4. B gets 3. 35 ways.
  d) A gets 3. B gets 4. 35 ways.
  e) A gets 2. B gets 5. 21 ways.
  f) A gets 1. B gets 6. 7 ways.
  
  Total = 126 ways to divide 7 items between A & B.
  Total number of ways for case 2 = 126*3 = 378 (Either A, or B, or C don't get anything).
  
  Answer = Case 1 + Case 2 = 3 + 378 = 381
• 8 years agoHelpfull: Yes(32) No(9)
• assume there are three persons A,B ,C
  now consider following two cases
  1.
  A do not get anything
  in this case 7 objects can be distributed between B and C as follows
  B C
  1 6 - total ways are 7!/1!*6!=7
  2 5 -total ways are 7!/2!*5!=21
  3 4 - total ways are 7!/3!*4!=35
  4 3 - total ways are 7!/4!*3!=35
  5 2 - total ways are 7!/5!*2!=21
  6 1 - total ways are 7!/6!*1!=7
  hence total ways of distributing objects between B and C = 126
  similarly for (A c)and(AB) each =126
  CASE2-
  two persons do not get anything
  in thi case there will be three waya these are
  007 ,070 ,700
  combining both the cases
  total no of ways =126*3+3=381
• 8 years agoHelpfull: Yes(17) No(0)
• assuming person C is empty handed
  1st object can be distributed to 2 person
  similarly 2nd,3rd,4th,5th,6th,7th
  no of ways =2*2*2*2*2*2*2
  now assuming B &a as empty handed
  total ways=3*2^7-3 =381
  we subtracted 3 ,becoz there is conidtion when only 1 person get the object is repeated twice i.e 6 instead of 3
• 8 years agoHelpfull: Yes(3) No(1)
• 36 is the answer
• 8 years agoHelpfull: Yes(1) No(6)
• 7c2+7c1=28 is answr
  
• 8 years agoHelpfull: Yes(1) No(3)
• 1st situation:
  when 2 of them don't get any object i.e. all 7 are given to 1 of them . so there can be 1 out of 3 which gets all so , 3 ways.
  
  2 situation :
  when one of them doesn't get any of the objects i.e. 7 objects are divided between two of them . Since these 7 are distinct objects so choices are also made that which object is given to which guy of the two.
  
  so,
  7c1+7c2+7c3+7c4+7c5+7c6=126 ways.
  Now this is the no. of ways in which objects are divided between 2 of them
  Now we can also select these two of the 3
  so we have 3 choices
  so final ans is:
  3*126+3=381.
  
  
• 8 years agoHelpfull: Yes(1) No(1)
• Case 1: Two guys don't get anything. All items go to 1 person. Hence, 3 ways.
  
  Case 2: 1 guy doesn't get anything. Division of 7 items among 2 guys. Lets say they are called A & B.
  
  a) A gets 6. B gets 1. 7 ways.
  b) A gets 5. B gets 2. 21 ways.
  c) A gets 4. B gets 3. 35 ways.
  d) A gets 3. B gets 4. 35 ways.
  e) A gets 2. B gets 5. 21 ways.
  f) A gets 1. B gets 6. 7 ways.
  
  Total = 126 ways to divide 7 items between A & B.
  Total number of ways for case 2 = 126*3 = 378 (Either A, or B, or C don't get anything).
  
  Answer = Case 1 + Case 2 = 3 + 378 = 381
• 4 years agoHelpfull: Yes(0) No(0)