A man takes a step forward with probability 0.4 and backward with probability 0.6. The probability that at the end of eleven steps he is one step away from the starting point.

Since the man is one step away from starting point mean that either
(i) man has taken 6 steps forward and 5 steps backward
(ii) man has taken 5 steps forward and 6 steps backward
Taking, movement 1 step forward as success and 1 step backward as failure
$\therefore$    p = Probability of success = 0.4
and  q = Probability of failure = 0.6
$\therefore$     Required Probability   $=P\{X=6orX=5\}$
                                            $$\begin{gathered} =P(X=6)+P(X=5) \\ {=}^{11}{C}_6{p}^6{q}^5{+}^{11}{C}_5{p}^5{q}^6 \\ {=}^{11}{C}_5(p^6{q}^5+p^5{q}^6)=\frac{\mathrm{11.10.9.8.7}}{\mathrm{1.2.3.4.5}}\{{\left(0.4\right)}^6{\left(0.6\right)}^5+{\left(0.4\right)}^5{\left(0.6\right)}^6\} \\ =\frac{\mathrm{11.10.9.8.7}}{\mathrm{1.2.3.4.5}}{\left(0.24\right)}^5 \\ =0.37 \end{gathered}$$
Hence the required probability   $=0.37$