 Hello and welcome to the session. Let us discuss the following question. It says prove that summation r running from 0 to n 3 to the power r into n c r is equal to 4 to the power n. Now to solve this question we need to know the expansion of a plus b whole to the power n. It is equal to n c 0 a to the power n plus n c 1 a to the power n minus 1 plus n c 2 a to the power n minus 2 b to the power 2. So on the last term is n c n b to the power n. So this is the key idea. Let us now proceed on with the solution. We have to prove that 4 to the power n is equal to this expression. Now 4 can be written as 1 plus 3. Therefore 4 to the power n is equal to 1 plus 3 whole to the power n. Now we expand this using this expression. Here a is 1 and b is 3. So 4 to the power n which is equal to 1 plus 3 whole to the power n and this is equal to n c 0 1 to the power n plus n c 1 1 to the power n minus 1 into 3 plus n c 2 1 to the power n minus 2 into 3 to the power 2. So on the last term will be n c n into 3 to the power n. Now this is equal to n c 0 1 to the power n is 1 itself plus n c 1 into 3 plus 3 square into n c 2 plus 3 to the power 3 into n c 3. So on last term is 3 to the power n into n c n. And we can write this in the summation form. Summation are running from 0 to n 3 to the power r into n c r. Hence we have proved that summation are running from 0 to n 3 to the power r into n c r is equal to 4 to the power n. So this completes the mission. Hope you enjoyed this session. Goodbye and take care.