Sarvesh K Verma Solutions for Chapter: Permutations & Combinations, Exercise 5: Introductory Exercise

Find the value of ${}^{10}\mathrm{C}_1+{}^{10}\mathrm{C}_2+{}^{10}\mathrm{C}_3+\dots +{}^{10}\mathrm{C}_{10}$.

Find the value of ${}^{10}\mathrm{C}_2+{}^{10}\mathrm{C}_3+\dots +{}^{10}\mathrm{C}_{10}$

Find the value of ${}^{10}\mathrm{C}_1+{}^{10}\mathrm{C}_2+{}^{10}\mathrm{C}_3+\dots +{}^{10}\mathrm{C}_9$

Find the value of ${}^{10}\mathrm{C}_0+{}^{10}\mathrm{C}_2+{}^{10}\mathrm{C}_4+\dots .$

Find the value of $\left[\left({}^{10}\mathrm{C}_1+{}^{10}\mathrm{C}_3+{}^{10}\mathrm{C}_5+\dots \right)\right.$

$\times \left({}^{20}\mathrm{C}_1+{}^{20}\mathrm{C}_3+{}^{20}\mathrm{C}_5+\dots \right)\times \left({}^{30}\mathrm{C}_1+{}^{30}\mathrm{C}_3+{}^{30}\mathrm{C}_5+\dots \right)\left.\times \dots \times \left({}^{100}\mathrm{C}_1+{}^{100}\mathrm{C}_3+{}^{100}\mathrm{C}_5+\dots .\right)\right]$

Larry has a long weekend off for $4$ days that starts from Thursday and lasts till Sunday. She wants to attend some spiritual enrichment sessions that are held every day for four hours. In how many ways can she attend one or more sessions during this long weekend?

Madhulika is planning to join International School for languages, which imparts the professional courses on numerous languages under various programs. Program $1$ offers $1$ language, program $2$ offers $2$ languages, program $3$ offers $3$ languages, and so on. No two programs offer any common language. One can choose any number of languages from a certain program, but all the languages must be chosen from any one program only. Currently, there are total $10$ programs on offer with the program $10$ offering exactly $10$ languages. Madhulika is willing to enroll for exactly $4$ languages, find the number of ways in which she can study any $4$ languages.

Paratha Inc. is a chain of food joints, which specializes in various kinds of parathas - the Indian dish, usually stuffed with various kinds of vegetables cooked and smashed. It has total $210$ different parathas in its menu. The menu is designed in such a way that ${\mathrm{n}}^{\mathrm{th} }$ pack contains $\mathrm{n}$ parathas. Also, no two packs contain common parathas. However, you can choose any $\mathrm{r}$ parathas from any pack $(\mathrm{r}\le \mathrm{n})$, but not from different packs, for any particular order.

The parathas can be ordered online only through home delivery service, and only one order in a day is accepted from a particular customer. Shivesh Raina - my foodie friend- wants to order 3 different parathas. Then, on a particular day, how many different sets of parathas can he choose from?