Players are selected for Judo based on their body weights from the following 10 weight groups:
1. (48 kg – 52 kg)
2. (52 kg – 56 kg)
3. (56 kg – 60 kg)
4. (60 kg – 64 kg)
5. (64 kg – 68 kg)
6. (68 kg – 72 kg)
7. (72 kg – 76 kg)
8. (76 kg – 80 kg)
9. (80 kg – 84 kg)
10. (84 kg – 88 kg)

The average weight of the players after selecting one player from each group is 68 kg. If one of the players (named S) leaves the team, their average weight comes down to 66.5 kg.

If one of the new two players is from group 4 (60–64 kg), which group is the other player from?

The correct option is 10 (Group 10).

Let's break down the solution step-by-step:

Step 1: Calculate the initial total weight of the 10 players
Initially, 1 player is selected from each of the 10 weight groups, making a total of 10 players.
The average weight of these 10 players is given as 68 kg.

$\text{Initial Total Weight}=10\times 68=680\text{kg}$

Step 2: Calculate the total weight after player S leaves
When player S leaves, there are 9 players remaining, and their average weight becomes 66.5 kg.

$\text{New Total Weight (9 players)}=9\times 66.5=598.5\text{kg}$

Step 3: Calculate the weight of player S
The weight of player S is the difference between the initial total weight and the total weight of the remaining 9 players:

$\text{Weight of S}=680-598.5=81.5\text{kg}$

Step 4: Determine the group of the second new player
To restore the team average of 68 kg after adding 2 new players to the 9 remaining players (making a total of 11 players):

$\text{Required Total Weight for 11 players}=11\times 68=748\text{kg}$

The combined weight of the 2 new players is:

$\text{Combined weight of 2 new players}=748-598.5=149.5\text{kg}$

One of the new players comes from Group 4, which has a weight range of 60 kg to 64 kg.
Subtracting this range from the combined weight of 149.5 kg gives the possible weight range for the second player:

$\text{Minimum weight of 2nd player}=149.5-64=85.5\text{kg}$

$\text{Maximum weight of 2nd player}=149.5-60=89.5\text{kg}$

A weight between 85.5 kg and 89.5 kg falls directly into Group 10 (84 kg – 88 kg).

Therefore, the other player must be from group 10.