2023HSCQ10A group with 5 students and 3 teachers is to be arranged in a circle. In how many ways can this be done if no more than 2 students can sit together? A. 4!×3!4!\times3!4!×3! B. 5!×3!5!\times3!5!×3! C. D.
2023HSCQ11 (b)In how many different ways can all the letters of the word CONDOBOLIN be arranged in a line?2 marksShort answer
2023HSCQ12 (d)It is known that (nr)=(n−1r−1)+(n−1r)\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}(rn)=(r−1n−1)+(rn−1) for all integers such that 1≤r≤n−11\le r\le n-11≤r≤n−1. (Do NOT prove this.) Find ONE possible set of values for ppp and qqq such that (202280)+(202281)+(20231943)=(pq).\binom{2022}{80}+\binom{2022}{81}+\binom{2023}{1943}=\binom{p}{q}.(802022)+(8120222 marksShort answer