Ten identical pencils are to be given away to five different people. (a) In how many different ways can that be done? (b) In how many different ways can it be done so that every person receives at least one pencil?

Answer:(a) 1001 ways(b) 126 waysStep-by-step explanation:In the question,Number of Identical pencils = 10Number of different people = 5Now,We know that for distributing 10 identical pencils among 5 people we will have to make 4 partitions.So, using a generalised formula,Number of ways of distributing 'n' identical things among 'r' objects is given by,[tex]^{n+r-1}C_{r-1}[/tex]So, using the same here we get,[tex]Number\,of\,ways=^{n+r-1}C_{r-1}\\Number\,of\,ways=^{10+5-1}C_{5-1}\\Number\,of\,ways=^{14}C_{4}=\frac{14!}{10!4!}=\frac{14\times 13\times 12\times 11}{4\times 3\times 2}\\Number\,of\,ways=1001\,ways[/tex]Therefore, there are 1001 number of ways of doing that.(b).For every person to receive atleast one pencil. We can put the partition everywhere but leaving one case where we are placing the partition before 1st pencil.So possible number of selections for the partition can now be made from only 9 pencils.Therefore,Number of ways is given by,[tex]^{n-1}C_{r-1}[/tex]where, 'n' is the number of identical objects and 'r' is the number of different people.So, Number of ways are,[tex]^{10-1}C_{5-1}=^{9}C_{4}=\frac{9!}{4!5!}=\frac{9\times 8\times 7\times 6}{4\times 3\times 2}=126[/tex]Therefore, the number of ways of distributing the pencils are 126.