Choice B is correct. The perimeter of a triangle is the sum of the lengths of all three of its sides. It's given that the lengths of two sides of a triangle are $4$ centimeters and $6$ centimeters. Let $x$ represent the length, in centimeters, of the third side of this triangle. The sum of the lengths, in centimeters, of all three sides of the triangle can be represented by the expression $4 + 6 + x$ . Since it’s given that the perimeter of the triangle is $18$ centimeters, it follows that $4 + 6 + x = 18$ , or $10 + x = 18$ . Subtracting $10$ from both sides of this equation yields $x equals 8$ . Therefore, the length, in centimeters, of the third side of this triangle is $8$ .

Choice A is incorrect. If the length of the third side of this triangle were $2$ centimeters, the perimeter, in centimeters, of the triangle would be $4 + 6 + 2$ , or $12$ , not $18$ .

Choice C is incorrect. If the length of the third side of this triangle were $10$ centimeters, the perimeter, in centimeters, of the triangle would be $4 + 6 + 10$ , or $20$ , not $18$ .

Choice D is incorrect. If the length of the third side of this triangle were $24$ centimeters, the perimeter, in centimeters, of the triangle would be $4 + 6 + 24$ , or $34$ , not $18$ .