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13 Jan 2026, 11:28
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At a customer support center, 7 agents worked a particular evening shift. Exactly 28 customer tickets were assigned during the shift to the agents, and each ticket was assigned to exactly one agent. Did each agent receive at least 1 ticket assignment?
(1) Each agent was assigned a different number of tickets.
(2) The agent who was assigned the most tickets was assigned 7 tickets.
(1) Each agent was assigned a different number of tickets.
(2) The agent who was assigned the most tickets was assigned 7 tickets.
13 Jan 2026, 11:28
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Official Solution:
At a customer support center, 7 agents worked a particular evening shift. Exactly 28 customer tickets were assigned during the shift to the agents, and each ticket was assigned to exactly one agent. Did each agent receive at least 1 ticket assignment?
(1) Each agent was assigned a different number of tickets.
If the agents received different numbers of tickets, one possible distribution is {0, 1, 2, 3, 4, 5, 13}, which sums to 28 and has all counts distinct. In this case, one agent received 0 tickets. Another valid distribution is {1, 2, 3, 4, 5, 6, 7}. In this case, all agents received at least 1 ticket. Not sufficient.
(2) The agent who was assigned the most tickets was assigned 7 tickets.
We know the maximum is 7, but we do not know the remaining distribution of the 21 other tickets. It is possible that all other agents received at least 1 ticket (for example, {7, 5, 4, 4, 4, 2, 2}), and it is also possible that one agent received 0 tickets (for example, {7, 6, 6, 4, 3, 2, 0}). Not sufficient.
(1)+(2):
From (2), the highest ticket count is 7. From (1), all 7 ticket counts must be different.
If we assume that 0 tickets is allowed, the largest total we could form using seven distinct nonnegative integers with a maximum of 7 is {7, 6, 5, 4, 3, 2, 0}. This sums to 27, which is less than the required total of 28. Therefore, 0 cannot be one of the ticket counts.
So the only possible set of seven distinct ticket counts with maximum 7 that sums to 28 is {7, 6, 5, 4, 3, 2, 1}.
In this distribution, every agent receives at least 1 ticket. Sufficient.
Answer: C
At a customer support center, 7 agents worked a particular evening shift. Exactly 28 customer tickets were assigned during the shift to the agents, and each ticket was assigned to exactly one agent. Did each agent receive at least 1 ticket assignment?
(1) Each agent was assigned a different number of tickets.
If the agents received different numbers of tickets, one possible distribution is {0, 1, 2, 3, 4, 5, 13}, which sums to 28 and has all counts distinct. In this case, one agent received 0 tickets. Another valid distribution is {1, 2, 3, 4, 5, 6, 7}. In this case, all agents received at least 1 ticket. Not sufficient.
(2) The agent who was assigned the most tickets was assigned 7 tickets.
We know the maximum is 7, but we do not know the remaining distribution of the 21 other tickets. It is possible that all other agents received at least 1 ticket (for example, {7, 5, 4, 4, 4, 2, 2}), and it is also possible that one agent received 0 tickets (for example, {7, 6, 6, 4, 3, 2, 0}). Not sufficient.
(1)+(2):
From (2), the highest ticket count is 7. From (1), all 7 ticket counts must be different.
If we assume that 0 tickets is allowed, the largest total we could form using seven distinct nonnegative integers with a maximum of 7 is {7, 6, 5, 4, 3, 2, 0}. This sums to 27, which is less than the required total of 28. Therefore, 0 cannot be one of the ticket counts.
So the only possible set of seven distinct ticket counts with maximum 7 that sums to 28 is {7, 6, 5, 4, 3, 2, 1}.
In this distribution, every agent receives at least 1 ticket. Sufficient.
Answer: C
18 Jan 2026, 19:11
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Bunuel when it is stated each one gets a different number of tickets, should we count "0" i.e. one who did not get any ticket. Different number of tickets, I think, does not include "no ticket"
