There are 12 points in a plane of which 5 points are collinear. The number of lines obtained by joining these points in pairs is \( {}^{12}C_2 - {}^{5}C_2 \).
False
Three letters can be posted in five letterboxes in \( 3^5 \) ways.
False
In the permutations of \( n \) things, \( r \) taken together, the number of permutations in which \( m \) particular things occur together is \( n^{m} P_{r-m} \times {}^{r}P_{m} \).
False
In a steamer there are stalls for 12 animals, and there are horses, cows and calves (not less than 12 each) ready to be shipped. They can be loaded in \( 3^{12} \) ways.
True
If some or all of \( n \) objects are taken at a time, the number of combinations is \( 2^n - 1 \).
True
There will be only 24 selections containing at least one red ball out of a bag containing 4 red and 5 black balls, given that balls of the same colour are identical.
True
Eighteen guests are to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. The number of ways in which the seating arrangements can be made is \( \dfrac{11!}{5!6!} (9!)(9!) \).
True
A candidate is required to answer 7 questions out of 12 questions divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. He can choose the seven questions in 650 ways.
False
To fill 12 vacancies there are 25 candidates of which 5 are from scheduled castes. If 3 vacancies are reserved for scheduled caste candidates while the rest are open to all, the number of ways of selection is \( {}^{5}C_3 \times {}^{20}C_9 \).
False