On two parallel lines \(a\) and \(b\), the points \(A_1, A_2, \dots , A_m\) and \(B_1, B_2, \dots , B_n\) are chosen, respectively, and all of the segments of the form \(A_iB_j\), where \(1 \leq i \leq m\), \(1 \leq j \leq n\). How many intersection points will there be if it is known that no three of these segments intersect at one point?
Prove the validity of the following formula of Newton’s binom \[(x+y)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}y + \dots + \binom{n}{n}y^n.\]
How many six-digit numbers exist, for which each succeeding number is smaller than the previous one?
Think of a way to finish constructing Pascal’s triangle upward.
Calculate the following sums:
a) \(\binom{5}{0} + 2\binom{5}{1} + 2^2\binom{5}{2} + \dots +2^5\binom{5}{5}\);
b) \(\binom{n}{0} - \binom{n}{1} + \dots + (-1)^n\binom{n}{n}\);
c) \(\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n}\).
In the expansion of \((x + y)^n\), using the Newton binomial formula, the second term was 240, the third – 720, and the fourth – 1080. Find \(x\), \(y\) and \(n\).
Show that any natural number \(n\) can be uniquely represented in the form \(n = \binom{x}{1} + \binom{y}{2} + \binom{z}{3}\) where \(x, y, z\) are integers such that \(0 \leq x < y < z\), or \(0 = x = y < z\).
How many four-digit numbers can be made using the numbers 1, 2, 3, 4 and 5, if:
a) no digit is repeated more than once;
b) the repetition of digits is allowed;
c) the numbers should be odd and there should not be any repetition of digits?
Here is a fragment of the table, which is called the Leibniz triangle. Its properties are “analogous in the sense of the opposite” to the properties of Pascal’s triangle. The numbers on the boundary of the triangle are the inverses of consecutive natural numbers. Each number is equal to the sum of two numbers below it. Find the formula that connects the numbers from Pascal’s and Leibniz triangles.
Find the sums of the following series:
a) \({\frac {1} {1 \times 2}} + {\frac {1} {2 \times 3}} + {\frac {1} {3 \times 4}} + {\frac {1} {4 \times 5}} + \dots\);
b) \({\frac {1} {1 \times 2 \times 3}} + {\frac {1} {2 \times 3 \times 4}} + {\frac {1} {3 \times 4 \times 5}} + {\frac {1} {4 \times 5 \times 6}} + \dots\);
c) \({\frac {0!} {r!}} + {\frac {1!} {(r-1)!}} + {\frac {2!} {(r-2) !}} + {\frac {3!} {(r-3)!}} + \dots\) for \(r \geq 2\).