Question

A box contains 5 different red and 6 different white balls. In how many ways can 6 balls be selected, so that there are atleast two balls of each colour?
Or
If all letters of the word 'DALDA' be arranged as in a dictionary. What is the twenty fifth word?

Answer 1

The selection of 6 balls, consisting of atleast two balls of each colour from 5 red and 6 white balls can be made in the following ways:
Case I The selection of 2 red balls out of 5 and 4 white balls out of 6 can be done in ${}^5{C}_2{\times }^6{C}_4$ ways.
Case II The selectriono of 3 red balls out of 5 and 3 white balls out of 6 can be done in ${}^5{C}_3{\times }^6{C}_3$ ways.
Case III The selection of 4 red balls out of 5 and 2 white balls out of 6 can be done ${}^5{C}_4{\times }^6{C}_2$ ways.
Hence, the total number of ways to select the balls
${=}^5{c}_2{\times }^6{C}_4{+}^5{C}_3{\times }^6{C}_3{+}^5{C}_4{\times }^6{C}_2=10\times 15+10\times 20+5\times 15=150+200+75=425$
Or
In dictionary, the words at each stage are aranged in alphabetical order. Starting with the letter A and arranging the four letters, i.e., ADDL, we obtain, $\frac{4!}{2!}=12$
thus there are 12 words which starts with A. These are teh first 12 words,
Then, starting with D and arranging other four letters AADL in different ways, we obtain $\frac{4!}{2!}=12$ words.
Thus, there are 12 words, which start with D.
Thus, we have so far constructed (12 + 12 ) = 24 words.
hence, the 25th word is LAADD.