 Hi and how are you all today? The question says the coefficient of the r minus one-th r-th and r plus one-th terms in the expansion of x plus one raised to the power n are in ratio one is to three is to find. Find both n and r. Let us discuss. Now here the binomial expression given to us is x plus one raised to the power n. First of all we need to find the coefficient of r minus one-th term that will be nc r minus two right because here rb is one only so it will have no effect on the other. Further the coefficient of r-th term will be nc r minus one and similarly the coefficient of r plus one-th term that we are used to write is nc r. Now according to the question that is given to us it's given to us that nc r minus two is to nc r minus one is to nc r is in ratio one is to three is to five which can be written as c n r minus two divided by one is equal to c n comma r minus one divided by three is equal to c n comma r divided by five. Now if we take c n r minus two divided by one is equal to c n minus n comma r minus one divided by three we can find out the value of at least one thing that will be on cross multiplying that will be three c n minus n comma r minus two is equal to one multiplied by c n comma r minus one. Now here it can be written as three now c n minus r can be written as n factorial divided by r minus two factorial and n plus two minus r factorial and this will be written as n factorial divided by r minus one factorial n plus one minus r factorial further we have three further n factorial will get cancelled out from the numerator on both the sides and we have three divided by r minus two factorial n plus two minus r and this will be written as n plus one minus r factorial where we have expanded n plus two minus r factorial similarly here we have one divided by r minus one factorial can be written as r minus one r minus two factorial and here we have n plus one minus r factorial now on comparing both the sides you can see that we can cancel out r minus two factorial from both the sides and n plus one minus r factorial from both the sides and we are left with three divided by n plus two minus r is equal to one divided by r minus one again on cross multiplying we have three r minus three is equal to n plus two minus r that is this r will be taken to the left inside and this three will be taken to the right inside and we have four r is equal to n plus five let this be the first equation now again if we have the other two can be compared off we have C n r minus one divided by three is equal to C n comma r divided by five cross multiplying we have here five this can be written as now n factorial divided by r minus one factorial n plus one minus r factorial whereas it can be written as three n factorial divided by r factorial then n minus r factorial on simply fine we have in the numerator now five r minus one factorial now this can be written as n plus one minus r fact minus r multiplied by n minus r factorial and here we have three divided by r multiplied by r minus one factorial and minus r factorial let us cancel out and we left with five r is equal to three n plus three minus r that is on cross multiplying this can be written as that is three r eight r is equal to three n plus three now let this be the second equation now from the first equation we had four r is equal to n plus five that is on multiplying both the sides by two we have eight r is equal to two n plus five now on substituting this value from three in two we have in place of eight r we can write down two n plus five and here we can write down three n plus three let us take this in this side and listen to this side we have five minus three is equal to three n minus two n which is two is equal to I have two was multiplied on both the sides so this will be five not ten not five so ten minus three will give us seven so the value of n is equal to seven now on substituting the value in any of the equations above we have eight r is equal to two n plus ten n is seven we have r is equal to fourteen plus ten divided by eight that is twenty four divided by eight that gives us the value of r as three so our final answer is the value of n is equal to seven and r is equal to three so this ends the question that was given to us take care