Sheela has 7 friends (Yogesh, Nikhilesh, Shashank, Amol, Shainu, Vivek, Girish) to choose from for the party, and she can decide how many friends to take. She also has 10 pairs of sandals and 17 different dresses. She likes 7 pairs of sandals and 15 dresses which she wants to wear for the party. To find the total number of different ways she can go to the party, we consider:
- Choosing any number of friends from the 7 friends.
- Choosing 7 pairs of sandals from the 10.
- Choosing 15 dresses from the 17.
The number of ways to choose any number of friends from 7 is the sum of combinations from choosing 0 friends to all 7:
∑k=07(7k)=27=128\sum_{k=0}^{7}\binom{7}{k}=2^7=128k=0∑7(k7)=27=128
The number of ways to choose 7 pairs of sandals out of 10 is:
(107)=10!7!×3!=120\binom{10}{7}=\frac{10!}{7!\times 3!}=120(710)=7!×3!10!=120
The number of ways to choose 15 dresses out of 17 is:
(1715)=17!15!×2!=136\binom{17}{15}=\frac{17!}{15!\times 2!}=136(1517)=15!×2!17!=136
Thus, the total number of ways she can go to the party is:
128×120×136=2,088,960128\times 120\times 136=2,088,960128×120×136=2,088,960
So, Sheela can go to the party in 2,088,960 different ways considering her choices of friends, sandals, and dresses with her preferences included.
