Find the total number of ways in which six ‘+’ and four ‘−’ signs can be arranged in a line such that no two ‘−’ signs occur together.
Question:

Find the total number of ways in which six ‘+’ and four ‘−’ signs can be arranged in a line such that no two ‘−’ signs occur together.

Solution:

Six ‘t’ signs can be arranged in a row in $\frac{6 !}{6 !}=1$ way

Now, we are left with seven places in which four different things can be arranged in ${ }^{7} \mathrm{P}_{4}$ ways.

Since all the four ‘-‘ signs are identical, four ‘-‘ signs can be arranged in $\frac{{ }^{7} P_{4}}{4 !}$ ways, i.e. 35 ways.

Number of ways $=1 \times 35=35$

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