 Hello and welcome to the session. In this session, we will discuss properties of binomial coefficients. Now, the binomial expansion of 1 plus x will raise to power n is equal to nC0 plus nC1 into x plus nC2 into x square plus so n up to nCR into x raise to power r plus so n up to plus nCn minus 1 into x raise to power n minus 1 plus nCn into x raise to power n where n is a positive integer. Here nC0, nC1, nC2, nCR and so on up to nCn are called the binomial coefficients. And when n is known, that is the index of the binomial is known, then we can write these binomial coefficients as C0, C1 and so on up to these binomial coefficients as C0, C1 and so on up to C. Let us discuss the properties of these binomial coefficients. And the first property is the coefficients from the beginning, the terms which are equidistant from the beginning and the end equal coefficients. Now, let us put this property. Now, in the expansion of 1 plus x whole raise to power n which is equal to nC0 plus nC1 into x plus nC2 into x square plus so into x raise to power r plus so n up to minus 1 into x raise to power n minus 1 plus nCn into x raise to power n. Now, the total term, so the adorn expression has n plus 1 terms on the beginning coefficient of r plus 1 th term from the beginning plus 1 th n plus 1 the whole minus r plus 1 the whole terms before it. Therefore, r plus 1 th term from the beginning, 1 th term from the beginning is so the coefficient minus 1 th term from the beginning coefficient of r plus 1 th term from the end is equal to n factorial over r factorial into n minus r the whole factor. Therefore, the coefficient from the beginning of binomial coefficients of the binomial coefficients is equal to the expansion of 1 plus x whole raise to power n is equal to nC0 plus nC1 into x plus nC2 into x square raise to power n. Now, put 1 plus 1 that is 2 raise to power n is equal to dot plus C1 plus C2 plus Cn that is when n is the sum of the binomial coefficients is equal to 3 raise to power n. Now, as we know that C0 is equal to 1 and this equation is equal to 1 it will be 2 raise to power n is equal to 1 plus C1 plus C2 plus C3 plus C1 plus C3 plus Cn is equal to 2 raise. Now, let us discuss the next property which is in the binomial expansion of 1 plus x whole raise to power n the sum of the coefficients is equal to the coefficients even terms. Now, we have the binomial expansion of 1 plus x whole raise to power n is equal to nC0 plus nC1 into x plus nC2 into x square plus so on n into x raise to power n. Now, putting x is equal to 1 in this expansion we get 1 plus 1 that is 2 raise to power n is equal to nC0 plus nC1 plus nC2 plus so on the equation number 1 equal to minus 1 in the given expansion we get 0 is equal to nC0 minus nC1 plus nC2 minus nC3 plus so on raise to power n into nCn. Now, let this be equation number 2. Now, adding power n is equal to 2 into nC0 plus nC2 plus nC4 plus so on raise to power n is equal to 2 into C0 plus C2 plus C4 plus so on the whole. Now, subtracting from 1 we get 2 raise to power n is equal to 2 into nC1 plus nC3 plus nC5 plus so on the whole is equal to 2 into C1 plus C3 plus C2 C0 plus C2 plus C4 is equal to C1 plus C3 plus C power n over 2 is equal to C0 plus C2 plus C4 plus so on is equal to C1 plus C3 plus C5 which further implies minus 1 is equal to C0 plus C2 plus C4 plus so on is equal to C1 plus C3 plus C5 plus so on. But in the binomial expansion of 1 plus x whole raise to power n the sum of the coefficients is equal to the sum of the coefficients of the even terms n minus 1. Therefore, we can write that the sum of the coefficients of odd terms is equal to 2 raise to power n by 2 which is equal to 2 raise to power n minus 1. The coefficients of even terms is also 2 raise to power n by 2 which is equal to 2 raise to power n minus 1. Now let us discuss the next property which is summation minus 1 whole raise to power into nC r where r varies from 0 to n C0 minus C1 plus C2 minus C3 plus C4 minus C5 plus so on up to minus 1 whole raise to power n is equal to 0. We know the binomial expansion of 1 plus x whole raise to power n which is equal to C0 plus C2 x C3 x cube plus so on up to Cn into x raise to power n equal to minus 1 will be equal to 1 plus x whole raise to power n will be 0 which is equal to C0 minus C1 plus C2 minus C3 plus so on up to 1 whole raise to power n. Therefore, we have minus 1 whole raise to power r into nC r where r varies from 0 to n is equal to this property which is equal to n over r r the whole into r minus 2 is equal to now n factorial is equal to n into n minus 1 the whole over r factorial is equal to n factorial is equal to n minus 1 the whole minus r minus 1 the whole is equal to n over r the whole into n minus 1 factorial that is n minus 1 whole factorial minus 1 the whole minus r minus 1 the whole is complete whole factorial this is equal to n over r the whole into now this complete expression will be equal to n minus 1 C r minus 1 is equal to n over r the whole into n minus 1 C r minus 1 then is equal to minus 1 whole upon r minus 1 the whole into r the whole into 1 whole upon r minus 1 the whole into we have 1 so in this session we have learnt about properties of binomial coefficients so this completes our session but we don't have enjoyed the session.