AnswersComputer ScienceDiscrete Mathematicsshow that C(n+1,k)=C(n,k-1)+C(n,k)June 01, 2023Archangel Macsikashow that C(n+1,k)=C(n,k-1)+C(n,k)The Answer to the Questionis below this banner.Can't find a solution anywhere?NEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?Get the Answers Now!You will get a detailed answer to your question or assignment in the shortest time possible.Here's the Solution to this QuestionC_{n + 1}^k = \frac{{(n + 1)!}}{{k!(n + 1 - k)!}}Cn+1k=k!(n+1−k)!(n+1)!C_n^{k - 1} + C_n^k = \frac{{n!}}{{(k - 1)!(n - k + 1)!}} + \frac{{n!}}{{k!(n - k)!}} = \frac{{n!k}}{{k!(n - k + 1)!}} + \frac{{n!(n - k + 1)}}{{k!(n - k + 1)!}} = \frac{{n!k + n!(n - k + 1)}}{{k!(n - k + 1)!}} = \frac{{n!(k + n - k + 1)}}{{k!(n - k + 1)!}} = \frac{{n!(n + 1)}}{{k!(n - k + 1)!}} = \frac{{(n + 1)!}}{{k!(n + 1 - k)!}}Cnk−1+Cnk=(k−1)!(n−k+1)!n!+k!(n−k)!n!=k!(n−k+1)!n!k+k!(n−k+1)!n!(n−k+1)=k!(n−k+1)!n!k+n!(n−k+1)=k!(n−k+1)!n!(k+n−k+1)=k!(n−k+1)!n!(n+1)=k!(n+1−k)!(n+1)!\frac{{(n + 1)!}}{{k!(n + 1 - k)!}} = \frac{{(n + 1)!}}{{k!(n + 1 - k)!}} \Rightarrow C_{n + 1}^k = C_n^{k - 1} + C_n^kk!(n+1−k)!(n+1)!=k!(n+1−k)!(n+1)!⇒Cn+1k=Cnk−1+Cnk
