Arrange the following number of teams in a sequential order from less to more with respect to the allotment of byes in a Knock-out Tournament:

(A) 05
(B) 14
(C) 32
(D) 12
(E) 63

The correct option is (C), (E), (B), (A), (D).

In a knock-out tournament, if the number of teams is not a power of 2 (such as 2, 4, 8, 16, 32, 64, etc.), some teams are given a "bye" in the first round to skip playing. The number of byes is calculated using the following formula:

$$\text{Number of Byes}=2^n-N$$

Where:
• N is the total number of teams.
• 2ⁿ is the next higher power of 2 greater than or equal to N.

Let us calculate the number of byes for each of the given numbers of teams:

1. For (A) 05 teams:
The next higher power of 2 greater than or equal to 5 is 8 (since 2³ = 8).
Number of byes = 8 - 5 = 3.

2. For (B) 14 teams:
The next higher power of 2 greater than or equal to 14 is 16 (since 2⁴ = 16).
Number of byes = 16 - 14 = 2.

3. For (C) 32 teams:
Since 32 is already a power of 2 (specifically 2⁵ = 32), the next higher power of 2 is 32.
Number of byes = 32 - 32 = 0.

4. For (D) 12 teams:
The next higher power of 2 greater than or equal to 12 is 16 (since 2⁴ = 16).
Number of byes = 16 - 12 = 4.

5. For (E) 63 teams:
The next higher power of 2 greater than or equal to 63 is 64 (since 2⁶ = 64).
Number of byes = 64 - 63 = 1.

Now, let us list the calculated byes for each case to arrange them from less to more:
• (C) 32 teams: 0 byes
• (E) 63 teams: 1 bye
• (B) 14 teams: 2 byes
• (A) 05 teams: 3 byes
• (D) 12 teams: 4 byes

Arranging these in sequential order from less to more byes gives:
(C) → (E) → (B) → (A) → (D)

Therefore, the correct sequential order is (C), (E), (B), (A), (D).