Question

Mohan has 8 friends. In how many ways can he invite one or more of them to dinner?
A. 255
B. 256
C. 15
D. 16

Answer 1

Hint: As we know combination determines the number of possible arrangements in a collection of items in which the order of the selection does not matter and hence, to solve the given equation of combination, apply the formula to get the combined terms and the formula is given as \[{}^n{C_{r - 1}} + {}^n{C_r} = {}^{n + 1}{C_r}\].

Formula used:
\[{}^n{C_{r - 1}} + {}^n{C_r} = {}^{n + 1}{C_r}\]
In which,
\[n\]= is the number of items.
\[r\]= number of items are taken at a time.
The number of combinations of ‘n’ different things taken ‘r’ at a time is denoted by \[{}^n{C_r}\], in which
\[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]

Complete step by step solution:
As Mohan selects one or more than one of his 8 friends. Hence,
\[{}^8{C_1} + {}^8{C_2} + {}^8{C_3} + ...... + {}^8{C_8}\]
\[\Rightarrow{2^8} - 1\]
\[\therefore 255\]
Therefore, Mohan can invite one or more of them to dinner in 255 ways.

Hence, option A is the right answer.

Additional information: Difference Between Combinations and Permutations:
In combinations, only selection is important whereas in the case of permutations, not only the selection but also the arrangement in a particular sequence is considered. In a combination, the order of selected objects is immaterial whereas in a permutation, the order is essential. To find the permutations of n different items, taken \['r'\] at a time: we first select r items from n items and then arrange them. So usually, the number of permutations exceeds the number of combinations.

Note: The key point to a combination is that there are no repetitions of objects allowed and order is not important to find a combination. In the same way we can find the number of Permutations of \['n'\] different things taken \['r'\] at a time is denoted by \[{}^n{P_r}\] , in which \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\].