Part Cn =47n =2Express your answer using thrce signlllcant (lgures....

Evaluate $_{n} C_{r}$ using the formula from this section. $$_{5} C_{2}$$

So given according to question, is Frankie C zero and fine and see, uh, so we know that N c is a big toe factory and my minus you play affect you. So here we are given as n is equipped with 20 is given zero. So value off going deep C zero would be Victorian 20 by 20 minus zero Maybe blade way. See you so their value would come as factory 20 divided by factorial 20 will reply one. So again the value off 20 c zero would come as one. So this is the uncertain question.

In this problem we are given that 10 C two Is nothing but 21. Now we know that N. C. R. There's nothing but in factorial by and in the door and minus off. Yeah so let's put that value here. This becomes in factorial by two factorial and -2 factories Equals to 21. In fact total can be written us and -2 factorial In two and -1 Into end divided by two factorial into and minus Two factorial which is equals to 21. This gets canceled with this this is in And 1 -1 is and square minus and equals to 21 into two wars. What do you do? So we get and squared minus and minus 42 equals to zero which is in squared minus seven plus six. In -42 equals to zero. This is N -7 into And plus six equals to zero. And it's either seven or -6. It cannot be a negative value And it's equals two seven. That's all.

We have the binomial expansion of one plus X. Whole power in S one plastic sole power. It is C. Zero. Let's see one X. Let's see. Two X square. Let's see. Three. Mhm. We have the binomial expansion of one. Politics hold power in a C note plus even explicit two X squared plus C. Three execute the so on C not is a short form of N C. Not that means basically cr means n C. R. We need to find the value of C zero square plus c one square. Basically we need to find a sum of the squares of the binomial coefficients. So what we need to do is we replace X with one by X. So it will be one place one by X. Whole power and will be C zero plus C. One by X plus C. Two by X square Plus it three by execute and so on and so forth. Now basically this is a polynomial. This is this is a polynomial in X. This is a polynomial in one by X. When I multiply both the pollen or meals. One is polynomial. Nx one is polynomial is with one by X. And uh definitely implicating terms like the C. Zero into C. Zero C zero square Stephen accent to see even by excess even square plus C. Two extra square into C. To buy extra square ec two square and so on. Apart from these help getting some some other towns also because see not gets multiplied with seven bags where you see not seven bags and some some blah blah blah. So different different terms you'll get. But we are interested in this. What is the specialty of this term is this is an independent term. There's no X in world in this term. This is called an independent term. So when you compare two expressions, obviously I should compare the independent town. So that that means our aim is to collect the independent town here. Also because the independent term in this should be same as this independent town. So what we do is we'll simplify this so we can write this as one plus six whole parent into one classics, whole parent divided by X parent. So basically it is one plus X. Whole power to in divided by expiring numerous is again in new binomial series, it is to N C zero. Let's to N C one X plus two N C two X square and so on. In the middle there is a two N C N X parent plus so on, divided by experience. Now, what will be the independent term in this particular expansion? Obviously this one because this gets divided with experience. That expiring gets canceled. So that means to n C N is free from X. It is actually alone, whereas this will be not alone because they involve the multiplication with the variable X. The answer is C. North square perceiving squares. The conclusion see North Square Perceiving Square. So on C n square will be the independent term that is to n C n two N C. And is the final answer. But you can use the binomial formula do in factorial by in fact, oh, really? Do in factory? So the answer is to in factorial divided by N factorial, the whole square.