A physical quantity P has the dimensional formula [L^ -1 T^ -1 K^...

First, find the dimensional formulas of the given constants:

Stefan-Boltzmann constant ($\sigma$): From $E = \sigma A t T^4$, $[\sigma] = \dfrac{[\text{M}\text{L}^2\text{T}^{-2}]}{[\text{L}^2][\text{T}][\text{K}^4]} = [\text{M}\text{T}^{-3}\text{K}^{-4}]$

Boltzmann constant ($k$): From $E = k T$, $[k] = \dfrac{[\text{M}\text{L}^2\text{T}^{-2}]}{[\text{K}]} = [\text{M}\text{L}^2\text{T}^{-2}\text{K}^{-1}]$

Wien's displacement constant ($b$): From $\lambda_{\text{max}} T = b$, $[b] = [\text{L}][\text{K}] = [\text{L}\text{K}]$

Now, check the dimensions of the combination $\sigma k^{-1} b$:

$[\sigma k^{-1} b] = \dfrac{[\sigma][b]}{[k]}$

$[\sigma k^{-1} b] = \dfrac{[\text{M}\text{T}^{-3}\text{K}^{-4}][\text{L}\text{K}]}{[\text{M}\text{L}^2\text{T}^{-2}\text{K}^{-1}]}$

$[\sigma k^{-1} b] = \dfrac{[\text{M}\text{L}\text{T}^{-3}\text{K}^{-3}]}{[\text{M}\text{L}^2\text{T}^{-2}\text{K}^{-1}]}$

$[\sigma k^{-1} b] = [\text{L}^{-1}\text{T}^{-1}\text{K}^{-2}]$

This matches the dimensional formula of $P$.

Answer: $\sigma k^{-1} b$