The number of ways in which one can select six distinct integers from the set $\{1, 2, 3, \cdots, 49\}$, such that no two consecutive integers are selected, is
(A) $\binom{49}{6} - 5\binom{48}{5}$
(B) $\binom{43}{6}$
(C) $\binom{25}{6}$
(D) $\binom{44}{6}$

The number of ways in which one can select six distinct integers from the set $\{1, 2, 3, \cdots, 49\}$, such that no two consecutive integers are selected, is
(A) $\binom{49}{6} - 5\binom{48}{5}$
(B) $\binom{43}{6}$
(C) $\binom{25}{6}$
(D) $\binom{44}{6}$

The number of ways in which one can select six distinct integers from the set $\{ 1, 2, 3, \cdots, 49 \}$, such that no two consecutive integers are selected, is
(A) $\binom { 49 } { 6 } - 5 \binom { 48 } { 5 }$
(B) $\binom { 43 } { 6 }$
(C) $\binom { 25 } { 6 }$
(D) $\binom { 44 } { 6 }$

An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is $\_\_\_\_$

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:
(1) 5148
(2) 6084
(3) 4356
(4) 14950

Eight buildings are arranged in a row, numbered 1, 2, 3, 4, 5, 6, 7, 8 from left to right. A telecommunications company wants to select three of these buildings' rooftops to install telecommunications base stations. If base stations cannot be installed on two adjacent buildings to avoid signal interference, how many ways are there to select locations for the base stations if no base station is installed on building 3?
(1) 12
(2) 13
(3) 20
(4) 30
(5) 35

A table consisting of two rows and 7 cells is given in the figure.
Patterns are created by painting 4 cells of this table black.
How many different patterns are there such that each row has at least one painted cell?
A) 26
B) 28
C) 30
D) 32
E) 34

Melisa will stack 10 circular cardboards with radii $1\text{ cm}, 2\text{ cm}, 3\text{ cm}, \cdots, 10\text{ cm}$ on a table with their centers coinciding, where each has a different natural number radius.
In how many different ways can Melisa arrange them so that when viewed from above, exactly one of the circles is completely hidden?
A) 36 B) 40 C) 45 D) 48 E) 54