In how many ways can Paa,Naana,Joe,Sela and Idi sit at a circular table so

Question

a.Idi  never sit next to Naana

b.Idi always sit next to Naana

Answer 1

(a) Using name initials for the people, we can see that if I and N never sit together, there can only be one or two spaces between them. The other people have 6 ways they can be arranged. So if I and N are seated with one person between them there are 6 arrangements for the other 3 people, and if I and N are seated with two people between them there are another 6 arrangements for the others. Remember also that they are sitting at a round table, so if I and N have one person between them clockwise, they automatically have two people between them anticlockwise. This makes 12 different ways altogether so that I and N are not sitting together:

IJNPS, IJNSP, IPNJS, IPNSJ, ISNPJ, ISNJP, IJPNS, IJSNP, IPJNS, IPSNJ, ISPNJ, ISJNP all fix the position of I as the starting point in the circle. This covers all cyclic arrangements. Note that there are 8 occurrences of I next to each of J, P and S. Similarly for N.

(b) If IN are together and there are 6 arrangements for the others, IN can be sandwiched in 3 places between the 3 other people, making 18 arrangements in all, or so it appears, because the cyclic arrangement gives us duplicates, which I've struck through:

JINPS, JPINS, JPSIN, JINSP, JSINP, JSPIN, PINJS, PJINS, PJSIN,

PINSJ, PSINJ, PSJIN, SINPJ, SPINJ, SPJIN, SINJP, SJINP, SJPIN.

We can therefore remove the struck through arrangements as duplicates, leaving 6 so far.

However, we can reverse the seating of IN to NI:

JNIPS, JPNIS, JPSNI, JNISP, JSNIP, JSPNI.

This gives 12 cyclic arrangements for I and N sitting together. Note that there are 4 occurrences of I and N next to each of J, P and S.

Comments

Is there any Mathematics formula for this kind questions

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We actually used one. The number of ways of arranging 3 different objects=3!=6. But remember that mathematics is a logical thinking process. Sometimes a formula is used (like solving quadratic equations, for example) but it's the logical process that matters. Breaking a problem down to bare essentials and simplifying the problem is part of the process. Looking at the problem in different ways is another process. Confirming your solution by using different approaches helps to build confidence that your solution is correct. Looking for patterns and making models and pictures also helps. It's all part of the mathematical process, rather than just looking for a formula to solve the problem.

Einstein failed mathematics when he was younger, but his contribution to mathematics evolved from thinking deeply about things, like, for example, riding on a beam of light. He needed assistance from other mathematicians to express his wonderful ideas in mathematical terms. So he didn't use formulas––they came later.