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

Let a step forward be a success and the step backward be a failure.
\(\therefore \quad\) Probability of success \(=\mathrm{p}=0.4\), and Probability of failure \(=q=0.6\)
Now, in 11 steps number of successes \(=6\), number of failure \(=5\)
OR
number of successes \(=5\), number of failures \(=6\)
Required probability \(={ }^{11} \mathrm{C}_6 \mathrm{p}^6 \mathrm{q}^5+{ }^{11} \mathrm{C}_5 \mathrm{p}^5 \mathrm{q}^6\).
\(\begin{aligned}
& =\frac{11 !}{6 ! 5 !} p^6 q^5+\frac{11 !}{5 ! 6 !} p^5 q^6 \\
& ={ }^{11} C_6 p^5 q^5(p+q) \\
& ={ }^{11} C_6(0-4)^5(0-6)^5(1) \\
& ={ }^{11} C_6(0 \cdot 4)^5(0 \cdot 6)^5 \\
& ={ }^{11} C_6(0 \cdot 24)^5
\end{aligned}\)