A shipping clerk has to weigh 6 distinct packets. He weighs them four at a time, weighting all the possible combinations of the packets from the six. The average weight of all the weighing combinations is found to be 500 g. What is the combined weight of all the six packets?

Answer

To find the combined weight of all six packets given that the average weight of all combinations of four packets is 500 grams, we can follow these steps:

### Step 1: Understand the problem
We have 6 distinct packets, and we are weighing them in combinations of 4 packets at a time. The average weight of these combinations is given as 500 grams.

### Step 2: Calculate the number of combinations
The number of ways to choose 4 packets from 6 can be calculated using the combination formula:
\[ \text{Number of combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Where \( n \) is the total number of packets (6) and \( r \) is the number of packets chosen (4).
\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15 \]
So, there are 15 combinations of packets.

### Step 3: Calculate the total weight of all combinations
Since the average weight of these combinations is 500 grams, we can find the total weight of all combinations:
\[ \text{Total weight of all combinations} = \text{Average weight} \times \text{Number of combinations} \]
\[ \text{Total weight of all combinations} = 500 \, \text{g} \times 15 = 7500 \, \text{g} \]

### Step 4: Relate total weight of combinations to total weight of packets
Each combination of 4 packets contributes to the total weight. Since each packet is included in multiple combinations, we need to determine how many times each packet is included in the combinations.

Each packet is included in:
\[ \text{Number of combinations including one packet} = \binom{5}{3} = 10 \]
This means each packet is included in 10 combinations.

### Step 5: Calculate the total weight of all packets
Let \( W \) be the combined weight of all 6 packets. Since each packet appears in 10 combinations, the total weight of all combinations can also be expressed as:
\[ \text{Total weight of all combinations} = 10W \]
Setting the two expressions for total weight equal gives:
\[ 10W = 7500 \, \text{g} \]
\[ W = \frac{7500 \, \text{g}}{10} = 750 \, \text{g} \]

### Step 6: Find the combined weight of all packets
Since \( W \) is the total weight of all 6 packets, we multiply by the number of packets:
\[ \text{Combined weight of all packets} = 750 \, \text{g} \]

### Final Answer
The combined weight of all six packets is **750 grams**.

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