 Hi and how are you all today? I am Priyanka and let us discuss this question. It says in the expansion of 1 plus a raised to the power m plus n prove that coefficients of a raised to the power m and a raised to the power n are equal. Now to proceed on with the solution we should be well versed with the expansion of a plus b raised to the power n. Here the r plus one-eth term is written as n c r a raised to the power n minus r b raised to the power r isn't it? Now the expansion of this binomial expansion is the key idea of our question. Let us proceed on with our solution. Now here we are given the binomial expansion for 1 plus a raised to the power m plus n. Now if you compare it with the key idea that we just discussed, we can find out the coefficient of a raised to the power m by knowing first of all m plus one-eth term that will be equal to m plus n as it is the n of the expression given to us c r over here is m a raised to the power here a is one n minus r then b raised to the power that is a over here r that is discussed as m. So here the b given to us is a further m plus n c m one raised to the power n as m will get cancelled out a raised to the power m which can further be written as m plus n c m m raised to the power n will give us one which can be neglected over here a raised to the power n. Now with whatever expansion that we have done we can know that this will be the coefficient of a raised to the power m hence here we can write down it as m plus n c m let this be the first equation. Similarly we need to find out the coefficient of a raised to the power n following the same procedure here we will find out t n plus one-eth term that will be m plus n c n as r is n over here a raised to the power n minus r n is m plus n and r is n a raised to the power n solving it these two terms will get cancelled further it can be written as m plus n c n one raised to the power m will be one only multiplied by a raised to the power n so here now here the coefficient of a raised to the power n will be m plus n c n isn't it so here we can write down it as m plus n c n let this be the second equation further now here we need to prove was that the coefficient of a raised to the power m is equal to coefficient of a raised to the power m this was the thing which was needed to be proved now in the above from one and two we can write down the coefficient of a raised to the power m that is m plus n c m m plus n c n further now we know that we can write this as m plus n factorial divided by m factorial n factorial similarly this can be written as m plus n factorial divided by m factorial n factorial so this proves that the coefficient of a raised to the power m and coefficient of a raised to the power n are equal so this ends our session have we proved the thing that was given to us I hope you enjoyed this session do remember the general term for r plus one-th term and you will be able to prove anything that you want I hope you enjoy take care