The structure of $B_4H_{10}$ can be analyzed by determining its styx number. The styx notation is a special representation used to describe the structure of borane compounds. It is denoted by four numbers: \((n+x+y+z)\), where:

• n - Number of boron atoms that are involved in three-center bonding with two hydrogen atoms.
• x - Number of boron atoms involved in two-center bonding with hydrogen atoms.
• y - Number of direct B–B bonds.
• z - Number of three-center B-H-B bonds.

Let us examine the structure provided:

Given the structure of $B_4H_{10}$:

1. All four boron atoms are involved in three-center B-H-B bonding.
2. Two boron-boron bonds are present in the compound.
3. Two three-center bridges (B-H-B) are present, involving hydrogens bridging between two borons.

Therefore, analyzing the structure, the styx numbers for $B_4H_{10}$ are:

• n = 4 (because all borons participate in three-center formation),
• x = 0 (no additional direct B-H two-center bonds),
• y = 1 (one direct B-B bond exists),
• z = 2 (two three-center B-H-B bonds).

Hence, the styx number is \(4012\).

Therefore, the correct answer is 4012.