 Hi, and how are you all today? I'm Priyanka. The question says, prove that the coefficient of x raised to the power n in 1 plus x raised to the power 2n is twice the coefficient of x raised to the power n in 1 plus x raised to the power 2n minus 1. Before proceeding on with the solution, we should be well versed with the general term in the expansion of 1 plus x raised to the power 2n. That is, if we have to find the 1 plus rth, r plus one-th term, it will be equal to c 2n comma r x raised to the power r. Now, if we put r is equal to n, we have n plus 1 is equal to c 2n. Now, in the case of r, we have n. So, we have this. Now, the coefficient of x raised to the power n above we have c 2n comma n. Right? Let this be the first equation. Similarly, for 1 plus x raised to the power 2n minus 1 the coefficient of x raised to the power n will be what? It will be equal to c 2n minus 1 comma n. Now, according to the question, let this be the second equation. According to the question given to us, we have to prove 2n comma n is equal to twice the coefficient of x raised to the power n in 1 plus x raised to the power 2n minus 1. That is c 2n minus 1 comma n. Now, we need to prove this thing. So, we have 2n factorial divided by n factorial 2n minus n factorial. And here it can be written as 2 multiplied by 2n minus 1 factorial divided by n factorial in bracket 2n minus 1 minus n factorial. Solving, we have 2n factorial divided by n factorial, 2n minus n factorial will be n factorial itself. It is equal to 2 multiplied by 2n minus 1 factorial divided by n factorial. And here it can be written as n minus 1 factorial. As on subtracting n from 2n, we are left with n and then we have minus 1. Further, 2n factorial divided by n factorial n factorial is equal to 2 multiplied by 2n minus 1 factorial divided by n factorial n minus 1 factorial. Now here, on multiplying the numerator, the denominator by n on RHS, we have in LHS, we have 2n factorial divided by n factorial n factorial. Now on this, we have 2n 2n minus 1 factorial divided by n factorial n minus 1 factorial. That is, now here, this can be written as 2n factorial only. This in the numerator can be written as 2n factorial n factorial. Now n minus 1 factorial is the expansion of n factorial only. So LHS will be equal to RHS and hence, we have proved the required given thing. This ends the question. Take care. .