Question

In how many ways can 7 plus $\left( + \right)$ and 5 minus $\left( - \right)$ signs be arranged in a row so that two minus $\left( - \right)$ signs are together?

Answer 1

In this question, we are given 7 plus $\left( + \right)$ and 5 minus $\left( - \right)$ signs and we have been asked to arrange these signs in a way that two minus $\left( - \right)$ signs are together. Start by arranging that sign which is more in number. Place them in such a way that there is always some place left between two signs to place the sign which is less in number. Tie two minus signs together as two signs should be together. After you have placed the signs, count the number of places you have for the other signs. Then place the signs in lesser numbers on those places.

Formula used: $^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$

Complete step-by-step solution:
We are given 7 plus $\left( + \right)$ and 5 minus $\left( - \right)$ signs and we have been asked to arrange them in such a way that two minus $\left( - \right)$ signs are together. For this purpose, we will tie two minus signs together. Since they all are the same, it won’t matter which two are tied together. Then we will consider the 2 tied signs to be 1 single sign and then we will have 4 minus $\left( - \right)$ signs.
Now, we will place the signs which are larger in number first, that is we will place plus signs in such a way that there is space for one minus sign to be placed between 2 plus signs.
\Rightarrow \\_\\_ + \\_\\_ + \\_\\_ + \\_\\_ + \\_\\_ + \\_\\_ + \\_\\_ + \\_\\_
This is what it will look like once we have placed the plus $\left( + \right)$ signs together. Now, we have 8 blank spaces to put 4 minus $\left( - \right)$ signs together.
Now, we have to choose 4 places out of 8 first.
${ \Rightarrow ^8}{C_4} = \dfrac{{8!}}{{4! \times 4!}}$
On solving we will get,
${ \Rightarrow ^8}{C_4} = \dfrac{{8 \times 7 \times 6 \times 5 \times 4!}}{{4! \times 4 \times 3 \times 2 \times 1}} = 70$

Therefore, 7 plus $\left( + \right)$ and 5 minus $\left( - \right)$ signs can be arranged in a row in 70 ways such that two minus $\left( - \right)$ signs are together.

Note: We can observe that, in mathematics, a combination is a selection of items from a collection, such that the order of selection does not matter (unlike permutations). Permutations differ from combinations, which are selections of some members of a set regardless of order.
The number of combinations of n objects taken r at a time is determined by the following formula:
$^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$