In how many ways 7 positive and 5 negative signs can be arranged in a

1.	In how many ways 7 positive and 5 negative signs can be arranged in a row so that no two negative signs occur together?
Answer» Since there is no condition for positive (+) sign, fix them in a row. +++++++ There are 6 places in between each plus and one before and one before and one after these positive (+) sign. i.e., these are 8 places for negative (-) sign and 5(-) negative signs are there. ∴ These negative (-) signs can be placed in ⁸C₅ ways. = ⁸C₅ = \(\frac{8!}{3!\space5!}\) = \(\frac{8.7.6}{3.2.1} = 56\)

In how many ways 7 positive and 5 negative signs can be arranged in a row so that no two negative signs occur together?

Answer»

Since there is no condition for positive (+) sign, fix them in a row.

+++++++

There are 6 places in between each plus and one before and one before and one after these positive (+) sign.

i.e., these are 8 places for negative (-) sign and 5(-) negative signs are there.

∴ These negative (-) signs can be placed in ⁸C₅ ways.

= ⁸C₅

= \(\frac{8!}{3!\space5!}\)

= \(\frac{8.7.6}{3.2.1} = 56\)