**Selection of One or More Objects**

**Selection from Different Objects: **The number of ways of selecting one or more items from a group of n distinct item is 2ⁿ – 1.

**Example:** A man has 6 friends, in how many ways may he invite one or more of them to dinner?

**Solution**: The man has to select some or all of his 6 friends. So, required number of ways = 2⁶ – 1 = 63.

**Selection from Identical Objects:**

- The number of ways of selecting r items out of n identical items is 1.
- The total number of ways of selecting zero or more i.e. at least one item from a group of n identical items is (n + 1).
- The total number of selection of some or all out of p + q + r items where p are alike of one kind, q are alike of second kind and rest are alike of third kind is [(p + 1) (q + 1) (r + 1)].

**Selection from Identical and District Objects: **The total number of ways of selecting one or more items from p identical items of 1 kind of q identical items of second kind; r identical items of third kind and n different items is (p + 1) (q + 1) (r + 1) 2ⁿ – 1.

**Example:** Find the number of factors (excluding 1 and the expression itself) of the product of a⁷b⁴c^{3}def where a, b, c, d, e, f are all prime numbers.

**Solution:** The total number of factors of the product a⁷b⁴c^{3}def is equal to the number of ways of selecting at least one from seven a’s, four b’s, three c’s, one d’s, one e’s and one f’s. The number of such ways is (7 + 1) (4 + 1) (3 + 1) (2) (2) (2) – 1 = 1279.

But this includes the given product. Hence the required number of factors = 1279 – 1 = 1278.