State whether the statements in True or False.

If some or all of n objects are taken at a time, the number of combinations is 2ⁿ–1.

State whether the statements in True or False.

In the permutations of n things, r taken together, the number of permutations in which m particular things occur together is ^n–mP_r–m × ^rP_m.

State whether the statements in True or 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 3¹² ways.

State whether the statements in True or False.

There will be only 24 selections containing at least one red ball out of a bag containing 4 red and 5 black balls. It is being given that the balls of the same colour are identical.

State whether the statements in True or False.

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 other side of the table. The number of ways in which the seating arrangementscan be made is .

[Hint: After sending 4 on one side and 3 on the other side, we have to select out of 11; 5 on one side and 6 on the other. Now there are 9 on each side of the long table and each can be arranged in 9! ways.]