Choice B is correct. A linear equation in one variable has no solution if and only if the equation is false; that is, when there is no value of $x$ that produces a true statement. It's given that in the equation $- 3 x + 21 p x = 84$ , $p$ is a constant and the equation has no solution for $x$ . Therefore, the value of the constant $p$ is one that results in a false equation. Factoring out the common factor of $minus 3 x$ on the left-hand side of the given equation yields $- 3 x (1 - 7 p) = 84$ . Dividing both sides of this equation by $negative 3$ yields $x (1 - 7 p) = - 28$ . Dividing both sides of this equation by $left parenthesis 1 minus 7 p right parenthesis$ yields $x = \frac{- 28}{1 - 7 p}$ . This equation is false if and only if $1 - 7 p = 0$ . Adding $7 p$ to both sides of $1 - 7 p = 0$ yields $1 equals 7 p$ . Dividing both sides of this equation by $7$ yields $\frac{1}{7} = p$ . It follows that the equation $x = \frac{- 28}{1 - 7 p}$ is false if and only if $p = \frac{1}{7}$ . Therefore, the given equation has no solution if and only if the value of $p$ is $\frac{1}{7}$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.