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 S leaves the group and two new players join the group, their average weight increases to 68 kg. These players can NOT be from which groups?

Correct Option: 1 and 3

Step-by-Step Explanation:

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 68 kg.

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

2. Determine the weight of player S:
When player S leaves, 9 players remain, and their new average weight becomes 66.5 kg.

$\text{Total Weight of remaining 9 players}=9\times 66.5=598.5\text{ kg}$

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

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

3. Calculate the required combined weight of the two new players:
After S leaves (leaving 9 players), two new players join, bringing the total count to 11 players.
The average weight of these 11 players increases back to 68 kg.

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

Let the combined weight of the two new players be W_new.

$598.5+W_{\text{new}}=748$

$W_{\text{new}}=748-598.5=149.5\text{ kg}$

Thus, the two new players must have a combined weight of 149.5 kg.

4. Evaluate the weight ranges of the given options:
Let's find the minimum and maximum possible combined weight for each given group combination to see which combination cannot total 149.5 kg.

Option 1: Groups 1 and 3
Group 1 range: 48 kg – 52 kg
Group 3 range: 56 kg – 60 kg

$\text{Minimum combined weight}=48+56=104\text{ kg}$

$\text{Maximum combined weight}=52+60=112\text{ kg}$

Since the maximum possible combined weight is only 112 kg, players from Groups 1 and 3 CANNOT reach the required combined weight of 149.5 kg.

Checking other options for verification:

Both from group 7:
Group 7 range: 72 kg – 76 kg
Combined range: 144 kg – 152 kg (149.5 kg lies within this range).

Groups 4 and 10:
Group 4 range: 60 kg – 64 kg
Group 10 range: 84 kg – 88 kg
Combined range: 144 kg – 152 kg (149.5 kg lies within this range).

Groups 5 and 9:
Group 5 range: 64 kg – 68 kg
Group 9 range: 80 kg – 84 kg
Combined range: 144 kg – 152 kg (149.5 kg lies within this range).

Hence, the two new players cannot be from 1 and 3.